Solve Systems of Equations Using Elimination Method

The aim of this lesson is to show a method to solve a system of linear equations by adding or subtracting the equations.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

How Many Toys Are There in Each Box?

Maya is researching a toy factory that currently produces robot action figures and car toys.

These toys are sold to stores in boxes. The total weight of the content of each box is of

20

kilograms. Each robot weighs

\frac{4}{5}

kilogram and each car weighs

\frac{1}{1 0}

kilogram. If

x

is the number of robots and

y

the number of cars in a box, the given information can be expressed by a linear equation.

\frac{4}{5} x + \frac{1}{1 0} y = 20

Also, each robot and each car are sold to the stores for

$ 15

and

$ 5,

respectively. The factory makes

$ 500

for each box sold to a store. This information can also be written as an equation.

15 x + 5 y = 500

These equations can be combined to form a system of linear equations.

{\frac{4}{5} x + \frac{1}{1 0} y = 20 15 x + 5 y = 500

Solve this system of equations without graphing. How many robots and cars are there in each box?

Discussion

Rewriting Equations

Sometimes the equations of a system can be rewritten in a way that is easier to work with. As an example, consider the following system of linear equations.

{x + 2 y = 3 \frac{1}{2} x - y = - \frac{1}{2} (I) (II)

In Equation (II), the coefficient of the variable

x

is a fraction. It would be easier to solve the system if all the coefficients were integers. It is possible to rewrite the system using equivalent systems.

Rule

Equivalent Systems

In a system of equations, an equivalent system can be created by replacing one equation with the sum of two or more equations in the system or by replacing an equation with a multiple of itself. An equation can be also replaced by the sum of that equation and a multiple of another equation in the system.

Proof

The statement will be proved for linear equations. For non-linear equations, the proof is similar. Consider a system of linear equations.

{a x + b y = c d x + e y = f (I) (II)

Let

(x_{1}, y_{1})

be a solution to the system. Therefore, this ordered pair satisfies both equations simultaneously.

{a x_{1} + b y_{1} = c ✓ d x_{1} + e y_{1} = f ✓

Consider now the system formed by the following two equations.

Equation (I)
Equation (II) plus a multiple of Equation (I)

The system described above is shown below.

{a x + b y = c d x + e y + α (a x + b y) = f + α c

It is already known that

(x_{1}, y_{1})

is a solution to the first equation of this new system. It needs to be verified that the ordered pair is also a solution to the second equation.

{a x_{1} + b y_{1} = c ✓ d x_{1} + e y_{1} + α (a x_{1} + b y_{1}) = ? f + α c

Since

(x_{1}, y_{1})

is a solution to the original system, it is known that

a x_{1} + b y_{1} = c

and that

d x_{1} + e y_{1} = f .

These two expressions can be substituted into the second equation above.

d x_{1} + e y_{1} + α (a x_{1} + b y_{1}) = ? f + α c

SubstituteII

$a x_{1} + b y_{1} = c$ , $d x_{1} + e y_{1} = f$

f + α c = f + α c ✓

Therefore,

(x_{1}, y_{1})

is also a solution to the second equation of the new system. This means that this ordered pair is a solution to the system formed by the first equation of the original system and the sum of the first equation and a multiple of the second equation.

{a x_{1} + b y_{1} = c ✓ d x_{1} + e y_{1} + α (a x_{1} + b y_{1}) = f + α c ✓

Equivalent systems can be used to solve systems of equations. One method to do this is called the Elimination Method.

Discussion

Elimination Method

Given a system of two equations in two variables, replacing one equation with the sum of that equation and a multiple of the other equation produces an equivalent system. This fact is used to solve systems of equations by the Elimination Method. Consider an example system of linear equations.

{3 x + 2 y = 6 y = 2 x - 11 (I) (II)

To solve the system by using the Elimination Method, there are five steps to follow.

Write the Equations in the Same Form

First, all like terms must be gathered on the same sides of the equations. In Equation (I), the variable terms are on the same side of the equation. However, the variable terms are on both sides of the equations in Equation (II). Like terms can be gathered on the same sides of the equations by applying the Properties of Equality.

{3 x + 2 y = 6 y = 2 x - 11

SubEqn

$(II):$ $LHS - 2 x = RHS - 2 x$

{3 x + 2 y = 6 - 2 x + y = - 11

Multiply an Equation

Multiply one of the equations by a constant so that one of the variable terms of the resulting equation is equal to or is the opposite of the corresponding variable term in the other equation. In this case, multiplying Equation (II) by

- 2

will produce opposite coefficients for the

y -

variable.

- 2 (- 2 x + y) = - 2 (- 11) ⇕ 4 x - 2 y = 22

Both the original and the resulting equations have the same solutions because they are equivalent equations.

Add the New Equation and the Other Equation

After the rewrites, the system of equations looks the following way.

{3 x + 2 y = 6 4 x - 2 y = 22

Add these two equations by adding the right-hand sides together and the left-hand sides together. This way one variable will be eliminated.

3 x + 2 y + (4 x - 2 y) = 6 + 22

▼

Simplify

RemovePar

Remove parentheses

3 x + 2 y + 4 x - 2 y = 6 + 22

CommutativePropAdd

Commutative Property of Addition

3 x + 4 x + 2 y - 2 y = 6 + 22

AddSubTerms

Add and subtract terms

7 x = 28

Note that this step results in an equation in only one variable. This equation can be solved by dividing both sides by

7 .

7 x = 28

DivEqn

$LHS / 7 = RHS / 7$

\frac{7 x}{7} = \frac{2 8}{7}

CalcQuot

Calculate quotient

x = 4

Write an Equivalent System

Substitute the value for the solved equation in one variable for any of the equations of the system. This produces an equivalent system of equations. In this case, Equation (I) will be replaced.

{3 x + 2 y = 6 - 2 x + y = - 11 (I) (II) ⇕ {x = 4 - 2 x + y = - 11

Note that the first equation is the solution value of

x .

Solve the Equivalent System

To solve the new system, substitute the found value into the other equation. In this case,

4

will be substituted into Equation (II) for

x

to find the value of

y .

{x = 4 - 2 x + y = - 11

Substitute

$(II):$ $x = 4$

{x = 4 - 2 (4) + y = - 11

▼

(II):

Solve for

y

MultNegPos

$(II):$ $(- a) b = - a b$

{x = 4 - 8 + y = - 11

AddEqn

$(II):$ $LHS + 8 = RHS + 8$

{x = 4 y = - 3

In this system, the value of

y

- 3 .

Therefore, the solution to the system of equations, which is the point of intersection of the lines, is

(4, - 3),

x = 4,

y = - 3 .

If at any step of the method a true statement is found, then the lines represented by the equations of the system are coincidental. The system would be a dependent system and have infinitely many solutions. Conversely, if at any step a false statement is found, then the lines are parallel. In this case, the system would be inconsistent and have no solution.

Be aware that when using this method, it is possible to make some mistakes in any of the steps. Therefore, it is important to check if the obtained solution is actually a solution to the original system.

Pop Quiz

Checking if the Coordinate Pair is a Solution

For each system of linear equations, verify whether the coordinate pair is a solution.

Random Check if Coordinate Pair is a Solution to a System of Equations

Example

The Price of Gold

A vlogger that Diego likes to watch bought silver and gold to make Olympic-style medals. The vlogger will show the process of making the medals as a multi-video series.

The price of gold is about

$ 2000

per ounce, while the price of of silver is about

$ 25

per ounce. The vlogger spent a total of

$ 7075

in both silver and gold. Let

g

and

s

be the ounces of gold and silver, respectively, that the vlogger bought. Then, the given information can be modeled by a linear equation.

25 s + 2000 g = 7075

Diego notes that if he subtracts

50

times the number of ounces of gold from

5

times the number of ounces of silver, there would still be

65

ounces left. This can also be written as an equation.

5 s - 50 g = 65

A system of equations is obtained by combining these two equations.

{25 s + 2000 g = 7075 5 s - 50 g = 65 (I) (II)

a Solve the system of equations graphically.

b Solve the system of equations using the Substitution Method.

c Solve the system of equations using the Elimination Method.

d Interpret the system in terms of consistency and independence.

Answer

a See solution.

b See solution.

c See solution.

d The system is consistent and independent.

Hint

a First write the equations in slope-intercept form.

b Isolate one variable in an equation.

c Recall that equivalent systems have the same solution.

d A system of equations can be inconsistent, consistent dependent, or consistent independent.

Solution

a To solve the system graphically, first each equation should be written in slope-intercept form. The variable

s

will arbitrarily be considered as the dependent variable, and

g

will be the independent variable.

{25 s + 2000 g = 7075 5 s - 50 g = 65 (I) (II)

▼

(I), (II):

Write in slope-intercept form

DivEqn

$(I):$ $LHS / 25 = RHS / 25$

{\frac{2 5 s + 2 0 0 0 g}{2 5} = 283 5 s - 50 g = 65

WriteSumFrac

$(I):$ Write as a sum of fractions

{\frac{2 5 s}{2 5} + \frac{2 0 0 0 g}{2 5} = 283 5 s - 50 g = 65

DivEqn

$(II):$ $LHS / 5 = RHS / 5$

{\frac{2 5 s}{2 5} + \frac{2 0 0 0 g}{2 5} = 283 \frac{5 s - 5 0 g}{5} = 13

WriteDiffFrac

$(II):$ Write as a difference of fractions

{\frac{2 5 s}{2 5} + \frac{2 0 0 0 g}{2 5} = 283 \frac{5 s}{5} - \frac{5 0 g}{5} = 13

$(I), (II):$ $\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

{\frac{2 5}{2 5} s + \frac{2 0 0 0}{2 5} g = 283 \frac{5}{5} s - \frac{5 0}{5} g = 13

$(I), (II):$ Calculate quotient

{1 s + 80 g = 283 1 s - 10 g = 13

$(I), (II):$ Identity Property of Multiplication

{s + 80 g = 283 s - 10 g = 13

SubEqn

$(I):$ $LHS - 80 g = RHS - 80 g$

{s = - 80 g + 283 s - 10 g = 13

AddEqn

$(II):$ $LHS + 10 g = RHS + 10 g$

{s = - 80 g + 283 s = 10 g + 13

Now, both equations can be graphed on the same coordinate plane.

Looking at the graph, it can be seen that the solution is about $3$ ounces of gold and a bit more than $40$ ounces of silver. In this case, since the point of intersection is not a lattice point, finding the exact solution is not possible.

b To solve the system using the Substitution Method, first a variable should be isolated from one of the equations. It seems easier to isolate

s

on Equation (II).

{25 s + 2000 g = 7075 5 s - 50 g = 65 (I) (II)

▼

(II):

Solve for

s

DivEqn

$(II):$ $LHS / 5 = RHS / 5$

{25 s + 2000 g = 7075 \frac{5 s - 5 0 g}{5} = 13

WriteDiffFrac

$(II):$ Write as a difference of fractions

{25 s + 2000 g = 7075 \frac{5 s}{5} - \frac{5 0 g}{5} = 13

MovePartNumRight

$(II):$ $\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

{25 s + 2000 g = 7075 \frac{5}{5} s - \frac{5 0}{5} g = 13

CalcQuot

$(II):$ Calculate quotient

{25 s + 2000 g = 7075 1 s - 10 g = 13

IdPropMult

$(II):$ Identity Property of Multiplication

{25 s + 2000 g = 7075 s - 10 g = 13

AddEqn

$(II):$ $LHS + 10 g = RHS + 10 g$

{25 s + 2000 g = 7075 s = 10 g + 13

Substitute

$(I):$ $s = 10 g + 13$

{25 (10 g + 13) + 2000 g = 7075 s = 10 g + 13

▼

(I):

Solve for

g

Distr

$(I):$ Distribute $25$

{250 g + 325 + 2000 g = 7075 s = 10 g + 13

AddTerms

$(I):$ Add terms

{2250 g + 325 = 7075 s = 10 g + 13

SubEqn

$(I):$ $LHS - 325 = RHS - 325$

{2250 g = 6750 s = 10 g + 13

DivEqn

$(I):$ $LHS / 2250 = RHS / 2250$

{g = 3 s = 10 g + 13

The vlogger bought

3

ounces of gold. It is possible to calculate the number of ounces of silver he purchased by substituting this value for

s

into Equation (II).

{g = 3 s = 10 g + 13 (I) (II)

Substitute

$(II):$ $g = 3$

{g = 3 s = 10 (3) + 13

▼

(II):

Solve for

s

Multiply

$(II):$ Multiply

{g = 3 s = 30 + 13

AddTerms

$(II):$ Add terms

{g = 3 s = 43

Therefore, the vlogger has

43

ounces of silver.

c To solve this system using the Elimination Method, it is important to remember that equivalent systems share the same solution. Equation (I) can be divided by

5

so that the variable

s

has the same coefficient in both equations.

{25 s + 2000 g = 7075 5 s - 50 g = 65 (I) (II)

▼

(I):

Simplify

DivEqn

$(I):$ $LHS / 5 = RHS / 5$

{\frac{2 5 s + 2 0 0 0 g}{5} = 1415 5 s - 50 g = 65

WriteSumFrac

$(I):$ Write as a sum of fractions

{\frac{2 5 s}{5} + \frac{2 0 0 0 g}{5} = 1415 5 s - 50 g = 65

MovePartNumRight

$(I):$ $\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

{\frac{2 5}{5} s + \frac{2 0 0 0}{5} g = 1415 5 s - 50 g = 65

CalcQuot

$(I):$ Calculate quotient

{5 s + 400 g = 1415 5 s - 50 g = 65

Now, to continue solving using the Elimination Method, Equation (II) will be subtracted from Equation (I).

{5 s + 400 g = 1415 5 s - 50 g = 65 (I) (II)

SysEqnSub

$(I):$ Subtract $(II)$

{5 s + 400 g - (5 s - 50 g) = 1415 - 65 5 s - 50 g = 65

▼

(I):

Solve for

g

Distr

$(I):$ Distribute $- 1$

{5 s + 400 g - 5 s + 50 g = 1415 - 65 5 s - 50 g = 65

AddSubTerms

$(I):$ Add and subtract terms

{450 g = 1350 5 s - 50 g = 65

DivEqn

$(I):$ $LHS / 450 = RHS / 450$

{g = 3 5 s - 50 g = 65

The vlogger has

3

ounces of gold this time, too. This value can be substituted into Equation (II) to find out how many ounces of silver he has.

{g = 3 5 s - 50 g = 65 (I) (II)

Substitute

$(II):$ $g = 3$

{g = 3 5 s - 50 (3) = 65

▼

(II):

Solve for

s

Multiply

$(II):$ Multiply

{g = 3 5 s - 150 = 65

AddEqn

$(II):$ $LHS + 150 = RHS + 150$

{g = 3 5 s = 215

DivEqn

$(II):$ $LHS / 5 = RHS / 5$

{g = 3 s = 43

In this case, using the Elimination Method might be a little more intuitive than using the Substitution Method. Note that both of these methods are more reliable than solving the system graphically.

d To interpret the system in terms of the independence and consistency, these concepts should be reviewed.

Concept	Definition
Consistent System	A system of equations that has at least one solution.
Inconsistent System	A system of equations that has no solution.
Dependent System	A system of equations with infinitely many solutions.
Independent System	A system of equations that has exactly one solution.

Since the system has exactly one solution, it is both consistent and independent.

Example

Buying Equipment for the School Gym

As part of his student council duties, Tadeo is in charge of getting equipment for the school gym. At the moment, the gym needs new basketballs and volleyballs. Each basketball costs $$ 75$ and each volleyball costs $$ 50 .$ The gym's budget is $$ 1400 .$

Tadeo can only go to the store once to buy the equipment. Each basketball weighs

22

ounces and each volleyball weighs

10

ounces. The total weight of the purchased balls is

364

ounces. For Tadeo to be able to spend all the budget in one go, the number of basketballs

b

and volleyballs

v

need to satisfy the following system of linear equations.

{75 b + 50 v = 1400 22 b + 10 v = 364 (I) (II)

a Solve the system using the Elimination Method.

b Verify the answer from Part A.

c Interpret the system in terms of the consistency and independence.

Answer

a See solution.

b See solution.

c The system is consistent and independent.

Hint

a What is the easiest way for a variable to have the same coefficient in both equations?

b Substitute the values from Part A into the original system of equations.

c A system of equations can be inconsistent, consistent independent, or consistent dependent.

Solution

a To solve the system using the Elimination Method, the coefficients of one variable have to be same to eliminate the term by adding or subtracting. Looking at the system, it can be noted that dividing Equation (I) by

- 5

will achieve this purpose.

{75 b + 50 v = 1400 22 b + 10 v = 364 (I) (II)

▼

(I):

Simplify

DivEqn

$(I):$ $LHS / (- 5) = RHS / (- 5)$

{\frac{7 5 b + 5 0 v}{- 5} = - 280 22 b + 10 v = 364

WriteSumFrac

$(I):$ Write as a sum of fractions

{\frac{7 5 b}{- 5} + \frac{5 0 v}{- 5} = - 280 22 b + 10 v = 364

MovePartNumRight

$(I):$ $\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

{\frac{7 5}{- 5} b + \frac{5 0}{- 5} v = - 280 22 b + 10 v = 364

MoveNegDenomToFrac

$(I):$ Put minus sign in front of fraction

{- \frac{7 5}{5} b + (- \frac{5 0}{5} v) = - 280 22 b + 10 v = 364

AddNeg

$a + (- b) = a - b$

{- \frac{7 5}{5} b - \frac{5 0}{5} v = - 280 22 b + 10 v = 364

CalcQuot

$(I):$ Calculate quotient

{- 15 b - 10 v = - 280 22 b + 10 v = 364

Now,

v

has opposite coefficients. Therefore, by adding the equations, this variable will be eliminated.

{- 15 b - 10 v = - 280 22 b + 10 v = 364 (I) (II)

SysEqnAdd

$(I):$ Add $(II)$

{- 15 b - 10 v + (22 b + 10 v) = - 280 + 364 22 b + 10 v = 364

▼

(I):

Solve for

b

RemovePar

$(I):$ Remove parentheses

{- 15 b - 10 v + 22 b + 10 v = - 280 + 364 22 b + 10 v = 364

AddSubTerms

$(I):$ Add and subtract terms

{7 b = 84 22 b + 10 v = 364

DivEqn

$(I):$ $LHS / 7 = RHS / 7$

{b = 12 22 b + 10 v = 364

Since

b = 12,

Tadeo can buy and carry

12

basketballs. To find the number of volleyballs he can buy, this value will be substituted into Equation (II).

{b = 12 22 b + 10 v = 364 (I) (II)

Substitute

$(II):$ $b = 12$

{b = 12 22 (12) + 10 v = 364

▼

(II):

Solve for

v

Multiply

$(II):$ Multiply

{b = 12 264 + 10 v = 364

SubEqn

$(II):$ $LHS - 264 = RHS - 264$

{b = 12 10 v = 100

DivEqn

$(II):$ $LHS / 10 = RHS / 10$

{b = 12 v = 10

Therefore, Tadeo can buy and carry

12

basketballs and

10

volleyballs.

b To verify the solution, the values found in Part A need to be substituted into the system of equations.

	Equation (I)	Equation (II)
Equation	$75 b + 50 v = 1400$	$22 b + 10 v = 364$
Substitute	$75 (12) + 50 (10) = 1400$	$22 (12) + 10 (10) = 364$
Simplify	$1400 = 1400 ✓$	$364 = 364 ✓$

The values verify both equations of the system. Therefore, the solution is correct.

c To interpret the system in terms of independence and consistency, these concepts will be reviewed.

Concept	Definition
Consistent System	A system of equations that has at least one solution.
Inconsistent System	A system of equations that has no solution.
Dependent System	A system of equations with infinitely many solutions.
Independent System	A system of equations that has exactly one solution.

Since the system has exactly one solution, the system is both consistent and independent.

Example

Tools in a Toolbox

Ignacio is looking through his dad's toolbox to see if there are any duplicates that can be sold at a garage sale. He discovers that there are some extra hammers and wrenches.

In fact, the number of wrenches is equal to the difference between

13

and twice the number of hammers. He figures out that if he sells each hammer at

$ 7

and each wrench at

$ 3.50,

he will make a profit of

$ 45.50 .

h

and

w

are the number of hammers and wrenches, respectively, the information can be written as a system of equations.

{w = 13 - 2 h 7 h + 3.50 w = 45.50 (I) (II)

a Solve the system of equations using the Elimination Method.

b Interpret the system in terms of consistency and independence.

Answer

a See solution.

b The system is consistent and dependent.

Hint

a In Equation (I), group the variable terms on the same side.

b A system of equations can be inconsistent, consistent independent, or consistent dependent.

Solution

a To solve the system using the Elimination Method, all the variables should be on the same side of the equation in both equations. This can be achieved by adding

2 h

to both sides of Equation (I).

{w = 13 - 2 h 7 h + 3.50 w = 45.50 (I) (II) ⇕ {2 h + w = 13 7 h + 3.50 w = 45.50 (I) (II)

Now, a variable needs to have the same or opposite coefficients in both equations. This can be achieved by multiplying Equation (I) by

3.5 .

{2 h + w = 13 7 h + 3.5 w = 45.5 (I) (II)

MultEqn

$(I):$ $LHS \cdot 3.5 = RHS \cdot 3.5$

{(2 h + w) 3.5 = 45.5 7 h + 3.5 w = 45.5

Distr

$(I):$ Distribute $3.5$

{7 h + 3.5 w = 45.5 7 h + 3.5 w = 45.5

SysEqnSub

$(II):$ Subtract $(I)$

{7 h + 3.5 w = 45.5 7 h + 3.5 w - (7 h + 3.5 w) = 45.5 - 45.5

Distr

$(II):$ Distribute $- 1$

{7 h + 3.5 w = 45.5 7 h + 3.5 w - 7 h - 3.5 w = 45.5 - 45.5

SubTerms

$(II):$ Subtract terms

{7 h + 3.5 w = 45.5 0 = 0 ✓

By doing these operations, Equation (II) becomes an identity, or a true statement. This means that this system of equations has infinitely many solutions. Realizing this, Ignacio thinks that it is too big of a coincidence and decides to count the tools again and change the prices before going to the garage sale.

b To interpret the system in terms of consistency and independence, these concepts should be reviewed.

Concept	Definition
Consistent System	A system of equations that has at least one solution.
Inconsistent System	A system of equations that has no solution.
Dependent System	A system of equations with infinitely many solutions.
Independent System	A system of equations that has exactly one solution.

Since the system has infinitely many solutions, the system is consistent and dependent. In a system of linear equations, if the system is dependent, then the equations that make the system are equations of coincidental lines.

Example

The Lengths of Two Tracks

LaShay is on the track and field team at school. She has two tracks nearby her home where she trains for races. She is trying to remember the lengths of the tracks.

If LaShay completes

2

laps on the first track and

1

lap on the second track, then she runs

5

miles. On the other hand, if she completes

6

laps on the first track and

3

laps on the second track, then she runs

18

miles. Letting

x

be the length of the first track and

y

the length of the second track, this information can be written as a system of linear equations.

{2 x + y = 5 6 x + 3 y = 18 (I) (II)

a Solve the system of equations using the Elimination Method.

b Interpret the system in terms of consistency and independence.

Answer

a See solution.

b The system is inconsistent.

Hint

a Make the coefficients of a variable term the same in both equations.

b A system of equations can be inconsistent, consistent independent, or consistent dependent.

Solution

a To solve the system using the Elimination Method, a variable term first needs to have the same coefficient in both equations. To achieve this, the first equation will be multiplied by

3 .

{2 x + y = 5 6 x + 3 y = 18 (I) (II)

▼

(I):

Simplify

MultEqn

$(I):$ $LHS \cdot 3 = RHS \cdot 3$

{(2 x + y) 3 = 15 6 x + 3 y = 18

Distr

$(I):$ Distribute $3$

{6 x + 3 y = 15 6 x + 3 y = 18

The

y -

variable has the same coefficient in both equations. Now, Equation (II) can be subtracted from Equation (I).

{6 x + 3 y = 15 6 x + 3 y = 18 (I) (II)

SysEqnSub

$(I):$ Subtract $(II)$

{6 x + 3 y - (6 x + 3 y) = 15 - 18 6 x + 3 y = 18

▼

(I):

Simplify

Distr

$(I):$ Distribute $- 1$

{6 x + 3 y - 6 x - 3 y = 15 - 18 6 x + 3 y = 18

SubTerms

$(I):$ Subtract terms

{0 = - 3 \times 6 x + 3 y = 18

Equation (I) was reduced to a false statement. Because of this, the equation has no solution. Knowing this, LaShay realizes that she does not remember the lengths of the tracks and decides to properly measure the tracks to prepare her training plan.

b To interpret the system in terms of consistency and independence, these concepts should be reviewed.

Concept	Definition
Consistent System	A system of equations that has at least one solution.
Inconsistent System	A system of equations that has no solution.
Dependent System	A system of equations with infinitely many solutions.
Independent System	A system of equations that has exactly one solution.

Since the system has no solution, it is an inconsistent system.

Pop Quiz

Practice Solving Systems of Linear Equations

Solve the system of linear equations to find the values of $x$ and $y .$

Closure

Finding Out How Many Toys Are in Each Box

At the beginning of this lesson, a system of equations was introduced for the number of toys in a box. In both equations,

x

is the number of robots and

y

the number of cars.

{\frac{4}{5} x + \frac{1}{1 0} y = 20 15 x + 5 y = 500 (I) (II)

Solve this system without graphing. How many robots and cars are there in each box?

Hint

Multiply or divide an equation by a number so that a variable has the same coefficient in both equations.

Solution

This system of linear equations can be solved without graphing using the Elimination Method. To use this method, Equation (I) will be multiplied by

10

to get rid of the fractions, while Equation (II) will be divided by

5

to make the coefficients of the

y -

variable equal.

{\frac{4}{5} x + \frac{1}{1 0} y = 20 15 x + 5 y = 500 (I) (II)

▼

(I), (II):

Simplify

MultEqn

$(I):$ $LHS \cdot 10 = RHS \cdot 10$

{(\frac{4}{5} x + \frac{1}{1 0} y) 10 = 200 15 x + 5 y = 500

Distr

$(I):$ Distribute $10$

{\frac{4}{5} x (10) + \frac{1}{1 0} y (10) = 200 15 x + 5 y = 500

CommutativePropMult

$(I):$ Commutative Property of Multiplication

{\frac{4}{5} (10) x + \frac{1}{1 0} (10) y = 200 15 x + 5 y = 500

MoveRightFacToNum

$(I):$ $\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

{\frac{4 0}{5} x + \frac{1 0}{1 0} y = 200 15 x + 5 y = 500

CalcQuot

$(I):$ Calculate quotient

{8 x + 1 y = 200 15 x + 5 y = 500

DivEqn

$(II):$ $LHS / 5 = RHS / 5$

{8 x + 1 y = 200 \frac{1 5 x + 5 y}{5} = 100

WriteSumFrac

$(II):$ Write as a sum of fractions

{8 x + 1 y = 200 \frac{1 5 x}{5} + \frac{5 y}{5} = 100

MovePartNumRight

$(II):$ $\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

{8 x + 1 y = 200 \frac{1 5}{5} x + \frac{5}{5} y = 100

CalcQuot

$(II):$ Calculate quotient

{8 x + 1 y = 200 3 x + 1 y = 100

$(I), (II):$ Identity Property of Multiplication

{8 x + y = 200 3 x + y = 100

Now that the variable

y

has the same coefficient in both equations, Equation (II) will be subtracted from Equation (I). Then, Equation (I) will be solved for the

x -

variable.

{8 x + y = 200 3 x + y = 100 (I) (II)

SysEqnSub

$(I):$ Subtract $(II)$

{8 x + y - (3 x + y) = 200 - 100 3 x + y = 100

▼

(I):

Solve for

x

Distr

$(I):$ Distribute $- 1$

{8 x + y - 3 x - y = 200 - 100 3 x + y = 100

SubTerms

$(I):$ Subtract terms

{5 x = 100 3 x + y = 100

DivEqn

$(I):$ $LHS / 5 = RHS / 5$

{x = 20 3 x + y = 100

Therefore, there are

20

robots in each box. Now, the value of

x

will be substituted into Equation (II) to find the value of

y .

{x = 20 3 x + y = 100

Substitute

$(II):$ $x = 20$

{x = 20 3 (20) + y = 100

▼

(II):

Solve for

y

Multiply

$(II):$ Multiply

{x = 20 60 + y = 100

SubEqn

$(II):$ $LHS - 60 = RHS - 60$

{x = 20 y = 40

There are

40

cars in each box.

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Catch-Up and Review

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