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Systems of equations are used to relate the values of two or more variables. There are different methods of solving a system of equations. This lesson will present these methods and show how to use them.
### Catch-Up and Review

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

Challenge

Vincenzo is fascinated by all things related to space and astronauts. He spends a lot of his free time reading books and watching movies about space travel, distant galaxies, and rocket science.

Vincenzo counted that he has watched or read $27$ things related to space movies or books in total. The number of movies he has seen is $9$ more than the number of books he has read. What are the numbers of movies and books about space that Vincenzo had watched or read?{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.77001953125em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Number<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">of<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">movies<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["18"]}}

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.77001953125em;vertical-align:-0.009765625em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Number<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">of<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">books<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":null,"answer":{"text":["9"]}}

Discussion

There are several methods for solving a system of equations. One of the most popular methods is the Substitution Method.

Method

The Substitution Method is an algebraic method for finding the solutions of a system of equations. It consists of *substituting* an equivalent expression for a variable in one of the equations of the system. Consider, for example, the following system of linear equations.
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${y−4=2x9x+6=3y (I)(II) $

To solve the system by using the Substitution Method, there are four steps to follow.
1

Isolate One Variable in Any of the Equations

The first step is to isolate *any* variable in *any* of the equations. For simplicity, in this case, the $y-$variable will be isolated in Equation (I).

2

Substitute the Expression

Substitute the new expression for the variable in the equation where the variable was not isolated. In this case, $2x+4$ will be substituted for $y$ in Equation (II).
Now Equation (II) only has *one* variable, which is $x.$

3

Solve the Equation With One Variable

Solve the equation that contains only one variable. In this case, Equation (II) will be solved for $x.$
The value of the $x-$variable is $2.$

${y=2x+49x+6=3(2x+4) (I)(II) $

Distr

$(II):$ Distribute $3$

${y=2x+49x+6=6x+12 $

SubEqn

$(II):$ $LHS−6=RHS−6$

${y=2x+49x=6x+6 $

SubEqn

$(II):$ $LHS−6x=RHS−6x$

${y=2x+43x=6 $

DivEqn

$(II):$ $LHS/3=RHS/3$

${y=2x+4x=2 $

4

Substitute the Value of the Variable Into the Other Equation

Now that the value of one of the variables is known, it can be substituted into the equation that has not been considered yet. Here, $x=2$ will be substituted into Equation (I).
The value of the $y-$variable in this system is $8.$ Therefore, the solution to the system of equations, which is the point of intersection of the lines, is $(2,8)$ or $x=2,$ $y=8.$

Example

After reading another book about space, Vincenzo quickly fell asleep and dreamed that he was an astronaut spacewalking for the first time. What an amazing experience!
Vincenzo received a task to install some external parts to the spaceship. The number of parts $p$ he installed and the number of minutes $m$ he spent in the open space are related by a system of equations.
### Answer

### Hint

### Solution

Now, graph both equations on the same coordinate plane. To graph the first equation, start by plotting the $y-$intercept of $42.$ Next, use the slope of $-4$ to move $1$ unit to the right and $4$ units down, or $2$ units to the right and $2⋅4=8$ units down, to plot the second point.
The value of $m$ is found to be $9.$ Now it can be substituted in either of the original equations. Notice that $p$ is already isolated in the first equation, so it might be convenient to substitute the value of $m$ into this equation and evaluate $p.$
The solution to the system of equations is $p=6$ and $m=9.$

External credits: @catalyststuff

${p+4m=428p−5m=3 $

a Solve the system by graphing.

b Solve the system by substitution.

c Are the solutions the same? Which method of solving is more useful in this case and why?

a **Graph:**

**Solution:** $m=9,$ $p=6$

b $m=9,$ $p=6$

c The solutions are the same. In this case, the Substitution Method is more useful for a number of reasons.

a Rewrite the equations in slope-intercept form. Then use the slope and $y-$intercept to graph each equation.

b Isolate $p$ in the first equation and substitute the corresponding expression into the second equation to find $m.$ Then substitute the value of $m$ into the first equation and find $p.$

c Identify which method is shorter. Does either method require the equations to be in a specific form? Do they both result in finding the exact solutions every time?

a In order to solve the system of equations by graphing, both equations should be written in slope-intercept form.

$y=mx+b $

Rewrite both equations until they match this form. Notice that the first equation is already almost in this form — all that is left is to subtract $4m$ from both sides.
$p+4m−4m=42−4m⇕p=-4m+42 $

Rewrite the second equation similarly.
$8p−5m=3$

▼

Write in slope-intercept form

AddEqn

$LHS+5m=RHS+5m$

$8p=3+5m$

DivEqn

$LHS/8=RHS/8$

$p=83+5m $

WriteSumFrac

Write as a sum of fractions

$p=83 +85m $

MovePartNumRight

$ca⋅b =ca ⋅b$

$p=83 +85 m$

CommutativePropAdd

Commutative Property of Addition

$p=85 m+83 $

Draw a line through the two plotted points to get the graph of the first equation.

The second equation can be graphed by following the same process.
The solution of the system of equations is represented by the point of intersection of the lines. If the point of intersection lies on lattice lines or their intersections, the exact solution will be determined. Otherwise, only an estimate of the solution might be found.

The lines intersect at $(9,6).$ Therefore, $m=9$ and $p=6,$ which indicates that Vincenzo spent $9$ minutes spacewalking and installed $6$ parts on the spaceship.

b The system of equations will now be solved by using the Substitution Method. Start by isolating the variable $p$ and substituting the corresponding expression into the second equation.

${p+4m=428p−5m=3 (I)(II) $

SubEqn

$(I):$ $LHS−4m=RHS−4m$

${p=42−4m8p−5m=3 $

Substitute

$(II):$ $p=42−4m$

${p=42−4m8(42−4m)−5m=3 $

Distr

$(II):$ Distribute $8$

${p=42−4m336−32m−5m=3 $

SubTerm

$(II):$ Subtract term

${p=42−4m336−37m=3 $

SubEqn

$(II):$ $LHS−336=RHS−336$

${p=42−4m-37m=-333 $

DivEqn

$(II):$ $LHS/(-37)=RHS/(-37)$

${p=42−4mm=9 $

${p=42−4mm=9 $

Substitute

$(I):$ $m=9$

${p=42−4(9)m=9 $

Multiply

$(I):$ Multiply

${p=42−36m=9 $

SubTerm

$(I):$ Subtract term

${p=6m=9 $

c Both methods of solving the system of equations gave the same solution. Therefore, both methods of solving are correct.

$GraphingMethod↘ (9,6) SubstitutionMethod↙ $

However, in this case, the Substitution Method can be more convenient because it is shorter and gives the exact solution. By comparison, the graphing method requires the equations to be in slope-intercept form and does not always result in finding the exact solution.
Example

In his dreams, Vincenzo gets to travel to planets far far away. Traveling to two distant planets Lunaris and Exosia from Earth takes $137$ years and $680$ years, respectively.
### Answer

### Hint

### Solution

It was calculated that $ℓ$ equals $13.$ Next, substitute this value into either of the original equations and solve for the other variable $e.$ In this case, the first equation will be used since $e$ is already isolated on one side.
The solution to the system is $ℓ=13$ and $e=28.$
The equations both simplified into true statements, so the solution is indeed correct!
Similarly, rewrite Equation (II) in slope-intercept form.

The distances from Earth to Lunaris $ℓ$ and to Exosia $e$ are given by the following system of equations.

${2ℓ+2=eℓ+3e=97 $

a Solve the system by using the Substitution Method.

b Check the solution by substituting it into both equations of the system.

c Graph the system of equations and analyze the coordinates of the point of intersection.

a $ℓ=13$ and $e=28$

b See solution.

c **Graph:**

a Isolate one variable in one of the equations. Substitute the corresponding expression into the other equation to solve for the other variable.

b Substitute the solution from Part A into the system of equations and see if true statements are found.

c Rewrite each equation in slope-intercept form. Then, graph the equations using the $y-$intercepts and slopes.

a The system of equations will be solved by using the Substitution Method. Start by isolating one variable in one equation. Notice that $e$ is already isolated in the first equation.

${2ℓ+2=eℓ+3e=97 ⇕{e=2ℓ+2ℓ+3e=97 $

Substitute the corresponding expression into the other equation. Then, solve for the other variable. ${e=2ℓ+2ℓ=13 $

Substitute

$(I):$ $ℓ=13$

${e=2(13)+2ℓ=13 $

Multiply

$(I):$ Multiply

${e=26+2ℓ=13 $

SubTerm

$(I):$ Subtract term

${e=28ℓ=13 $

b To check the solution, substitute $13$ for $ℓ$ and $28$ for $e$ into the system of equations. If both equations result in true statements after simplification, the solution is correct.

$2ℓ+2=eℓ+3e=97 $

SubstituteII

$ℓ=13$, $e=28$

$2(13)+2=?2813+3(28)=?97 $

Multiply

Multiply

$26+2=?2813+84=?97 $

AddTerms

Add terms

$28=28✓97=97✓ $

c To solve the system of equations by graphing, start by rewriting both equations in slope-intercept form. Consider $ℓ$ as the $y-$variable and $e$ as the $x-$variable. Start with Equation (I).

$ℓ+3e=97⇓ℓ=-3e+97 $

Now, graph the equations using their $y-$intercepts and slopes. The point of intersection represents the solution.
The point of intersection lies on a lattice line where $e=28.$ However, it can be difficult to determine the exact value of $ℓ$ just by looking at the graph. It can have values from $11$ to $14.$ In Part A it was found that $ℓ$ is $13.$ The graph does support that value, so the solution is $(28,13).$

Discussion

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.
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${3x+2y=6y=2x−11 (I)(II) $

To solve the system by using the Elimination Method, there are five steps to follow.
1

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.

2

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(-2x+y)=-2(-11)⇕4x−2y=22 $

Both the original and the resulting equations have the same solutions because they are equivalent equations. 3

Add the New Equation and the Other Equation

After the rewrites, the system of equations looks the following way.
Note that this step results in an equation in only *one* variable. This equation can be solved by dividing both sides by $7.$

${3x+2y=64x−2y=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.
$3x+2y+(4x−2y)=6+22$

▼

Simplify

RemovePar

Remove parentheses

$3x+2y+4x−2y=6+22$

CommutativePropAdd

Commutative Property of Addition

$3x+4x+2y−2y=6+22$

AddSubTerms

Add and subtract terms

$7x=28$

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.

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

Note that the first equation is the solution value of $x.$ 5

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 $y.$
In this system, the value of $y$ is $-3.$ Therefore, the solution to the system of equations, which is the point of intersection of the lines, is $(4,-3),$ or $x=4,$ $y=-3.$

Example

Vincenzo and his team reached the planet Exosia and made a short stop there to refuel and repair their spaceship. The people of Exosia help Vincenzo and his crew make some modifications to their ship so they can travel at even greater speeds!
Their initial maximum speed $s$ and the improved intergalactic superspeed $n$ are related by the following system of equations.
### Answer

*<*

External credits: @catalyststuff

${2n−s=377s−4n=-19 $

a Solve this system by graphing.

b Solve the same system by elimination.

c Are the solutions the same?