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Algebra 1 View details

1. Solving Systems of Equations by Graphing

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Algebra 1 /

Chapter 5

Solving Systems of Equations by Graphing

Solving systems of equations by graphing is a fundamental concept in algebra. This lesson introduces learners to the methods of graphically solving systems of linear equations. By plotting these equations on a coordinate plane, one can determine the point where the lines intersect, which represents the solution to the system. Real-life scenarios, such as the number of students participating in high school sports and the sales of different car models, are used to demonstrate the practical applications of these mathematical techniques. Additionally, the lesson emphasizes the importance of understanding consistent, inconsistent, dependent, and independent systems. Graphing not only provides a visual representation but also aids in predicting trends and making informed decisions based on the data.

Lesson Settings & Tools

	13 Theory slides
	11 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Image Credits

Slide of 13

In this lesson, it will be shown how a graph can help find values of variables that satisfy two linear equations simultaneously.

Catch-Up and Review

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

Linear Equation
Slope-intercept form
Graphing linear equations in slope-intercept form
Literal Equations

Here are a few practice exercises before getting started with this lesson.

a Write the equation

- 6 x + 2 y = 12

in slope-intercept form.

b Consider the following linear graphs.

Which graph corresponds to

y = x - 1 ?

Challenge

Basketball and Soccer

The number of students in the US participating in high school basketball and soccer has steadily increased over the past few years.

The following table shows some information about this.

High School Sport	Number of Students Participating in $2015$ (Thousands)	Average Rate of Increase (Thousands per Year)
Basketball	$700$	$20$
Soccer	$400$	$40$

Consider the above table to answer the following questions.

a Write two equations to represent the situation.

b Graph the equations from Part A in the same coordinate plane. Use the graph to predict the approximate year when the number of students participating in these two sports will be the same.

Explore

Lines on the Same Plane

In the graph below, four lines and their corresponding linear equations can be seen on a coordinate plane. Investigate which lines intersect at one point, which lines intersect at infinitely many points, and which lines do not intersect at all.

Discussion

System of Equations

Two important concepts will be discussed here.

Concept

Equation In Two Variables

An equation in two variables is a mathematical relation between two equal quantities that involves two variables.

Solving an equation in two variables results in an ordered pair that makes the equation true.

Concept

System of Equations

A system of equations is a set of two or more equations involving the same variables. The solutions to a system of equations are values for these variables that satisfy all the equations simultaneously. A system of equations is usually written as a vertical list with a curly bracket on the left-hand side.

{2 x - 3 y = 1 3 x + y = 7

Graphically, solutions to systems of equations are the points where the graphs of the equations intersect. For this reason, these solutions are usually expressed as coordinates.

Systems of equations can be solved graphically or algebraically.

A system that contains only linear equations is called a linear system.

Discussion

Solving a System of Linear Equations Graphically

Solving a system of linear equations graphically means graphing the lines represented by the equations of the system and identifying the point of intersection. Consider an example system of equations.

{2 y = - 2 x + 8 x = y - 1

To solve the system of equations, three steps must be followed.

Write the Equations in Slope-Intercept Form

Start by writing the equations in slope-intercept form by isolating the

y -

variables. For the first linear equation, divide both sides by

2 .

For the second equation, add

1

to both sides.

{2 y = - 2 x + 8 x = y - 1 (I) (II)

▼

(I):

Solve for

y

DivEqn

$(I):$ $LHS / 2 = RHS / 2$

{y = \frac{- 2 x + 8}{2} x = y - 1

WriteSumFrac

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

{y = \frac{- 2 x}{2} + \frac{8}{2} x = y - 1

MovePartNumRight

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

{y = \frac{- 2}{2} x + \frac{8}{2} x = y - 1

MoveNegNumToFrac

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

{y = - \frac{2}{2} x + \frac{8}{2} x = y - 1

CalcQuot

$(I):$ Calculate quotient

{y = - 1 x + 4 x = y - 1

IdPropMult

$(I):$ Identity Property of Multiplication

{y = - x + 4 x = y - 1

▼

(II):

Solve for

y

AddEqn

$(II):$ $LHS + 1 = RHS + 1$

{y = - x + 4 x + 1 = y

RearrangeEqn

$(II):$ Rearrange equation

{y = - x + 4 y = x + 1

Graph the Lines

Now that the equations are both written in slope-intercept form, they can be graphed on the same coordinate plane.

Graphs of two lines using the slopes and y-intercepts

Identify the Point of Intersection

The point where the lines intersect is the solution to the system.

The lines appear to intersect at $(1.5, 2.5) .$ Therefore, this is the solution to the system — the value of $x$ is $1.5$ and the value of $y$ is $2.5 .$

Sometimes the point of intersection of the lines is not a lattice point. In this case, the solution found by solving the system of equations graphically is approximate.

Example

Modeling With a System of Linear Equations

Tearrik is throwing a party and bought some sodas and chips.

He bought

21

items and spent

$ 51 .

The cost of a soda is

$ 2

and the cost of a bag of chips is

$ 3 .

This can be modeled by a system of equations.

{x + y = 21 2 x + 3 y = 51 (I) (II)

Here,

x

is the number of sodas and

y

the number of bags of chips Tearrik bought. Solve the system by graphing and find how many sodas and bags of chips he bought.

Hint

Start by writing both linear equations in slope-intercept form.

Solution

To start, each equation in the system will be written in slope-intercept form. This means that the variable

y

will be isolated in both equations.

{x + y = 21 2 x + 3 y = 51 (I) (II)

▼

(I), (II):

Write in slope-intercept form

SubEqn

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

{y = - x + 21 2 x + 3 y = 51

SubEqn

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

{y = - x + 21 3 y = - 2 x + 51

DivEqn

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

{y = - x + 21 y = \frac{- 2 x + 5 1}{3}

WriteSumFrac

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

{y = - x + 21 y = \frac{- 2 x}{3} + \frac{5 1}{3}

MovePartNumRight

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

{y = - x + 21 y = \frac{- 2}{3} x + \frac{5 1}{3}

MoveNegNumToFrac

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

{y = - x + 21 y = - \frac{2}{3} x + \frac{5 1}{3}

CalcQuot

$(II):$ Calculate quotient

{y = - x + 21 y = - \frac{2}{3} x + 17

Now, the slope and the

y -

intercept of each line will be used to draw the graphs on the same coordinate plane. Since the number of items cannot be negative, only the first quadrant will be considered for the graph.

Finally, the point of intersection $P$ can be identified.

The point of intersection of the lines is $P (12, 9) .$ In the context of the situation, this means that Tearrik bought $12$ sodas and $9$ bags of chips.

Pop Quiz

Finding the Solution to a System of Equations by Graphing

The lines that correspond to a system's equations are graphed below. What is the solution to the system?

Discussion

Consistent and Independent Systems

The system of equations in the previous example had one solution. This leads to two important definitions.

Concept

Consistent System

A system of equations that has one or more solutions is called a consistent system. For example, consider the following linear systems.

\underline{Example System I} {y = x + 1 y = 2 x - 2 \underline{Example System II} {y = x - 1 2 y = 2 x - 2

To determine the number of solutions, each system can be graphed on a coordinate plane.

In the graph, it can be seen that the first system has exactly one solution. The second system has infinitely many solutions. Therefore, since both systems have one or more solutions, they are consistent systems.

Concept

Independent System

An independent system is a system of equations with exactly one solution. Consider the following linear system.

{y = x + 1 y = 2 x - 2

To state the number of solutions, both lines of the system can be graphed on the same coordinate plane.

Since the system has exactly one solution, it is an independent system.

Example

Finding the Solution and Classifying a System of Equations

At Tearrik's party, Ignacio found one of Tearrik's homework assignments. He does not want to do Tearrik's homework for him, but Ignacio decides to quiz himself using the assignment.

Help Ignacio determine whether the given systems of equations are consistent systems and whether they are independent systems.

{y = - 2 x + 6 4 x - 2 y = - 20

{x + y = 3 4 x + 4 y = 12

Hint

a Solve the system by graphing and determine the number of solutions.

b Solve the system by graphing and determine the number of solutions.

Solution

a The system will be solved graphically. To do so, both linear equations must to be expressed in slope-intercept form. In the first equation, the variable

y

is already isolated. Therefore, only the second equation needs to be rewritten.

{y = - 2 x + 6 4 x - 2 y = - 20 (I) (II)

▼

(II):

Write in slope-intercept form

SubEqn

$(II):$ $LHS - 4 x = RHS - 4 x$

{y = - 2 x + 6 - 2 y = - 4 x - 20

DivEqn

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

{y = - 2 x + 6 y = \frac{- 4 x - 2 0}{- 2}

WriteDiffFrac

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

{y = - 2 x + 6 y = \frac{- 4 x}{- 2} - \frac{2 0}{- 2}

DivNegNeg

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

{y = - 2 x + 6 y = \frac{4 x}{2} - \frac{2 0}{- 2}

MovePartNumRight

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

{y = - 2 x + 6 y = \frac{4}{2} x - \frac{2 0}{- 2}

MoveNegDenomToFrac

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

{y = - 2 x + 6 y = \frac{4}{2} x - (- \frac{2 0}{2})

SubNeg

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

{y = - 2 x + 6 y = \frac{4}{2} x + \frac{2 0}{2}

CalcQuot

$(II):$ Calculate quotient

{y = - 2 x + 6 y = 2 x + 10

Now that both linear equations are expressed in slope-intercept form, their slopes and

y -

intercepts can be used to draw the lines on the same coordinate plane.

The lines intersect at exactly one point.

Since the system has one or more solutions, it is a consistent system. Furthermore, since it has exactly one solution, it is also an independent system.

b Again, the system of equations will be solved graphically. To do so, both equations will be rewritten in slope-intercept form.

{x + y = 3 4 x + 4 y = 12 (I) (II)

▼

(I), (II):

Write in slope-intercept form

SubEqn

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

{y = - x + 3 4 x + 4 y = 12

SubEqn

$(II):$ $LHS - 4 x = RHS - 4 x$

{y = - x + 3 4 y = - 4 x + 12

DivEqn

$(II):$ $LHS / 4 = RHS / 4$

{y = - x + 3 y = \frac{- 4 x + 1 2}{4}

WriteSumFrac

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

{y = - x + 3 y = \frac{- 4 x}{4} + \frac{1 2}{4}

MoveNegNumToFrac

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

{y = - x + 3 y = - \frac{4 x}{4} + \frac{1 2}{4}

MovePartNumRight

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

{y = - x + 3 y = - \frac{4}{4} x + \frac{1 2}{4}

CalcQuot

$(II):$ Calculate quotient

{y = - x + 3 y = - 1 x + 3

IdPropMult

$(II):$ Identity Property of Multiplication

{y = - x + 3 y = - x + 3

Note that both equations are simplified to the same equation. It can be graphed by using its slope and its

y -

intercept.

The lines overlap each other. Therefore, they intersect at infinitely many points. Since the system has one or more solutions, it is a consistent system. However, since it does not have exactly one solution, it is not an independent system.

Discussion

Dependent and Inconsistent Systems

In the last example, one of the systems had infinitely many solutions. Systems of equations with infinitely many solutions have a special name.

Concept

Dependent System

A dependent system is a system of equations with infinitely many solutions. Consider the following linear system.

{2 y = 4 x - 4 y = 2 x - 2

To find the number of solutions, both lines of the system can be graphed on the same coordinate plane.

The lines overlap each other. Therefore, the system has infinitely many solutions. This means that this is a dependent system.

Furthermore, systems with no solutions also have a special name.

Concept

Inconsistent System

A system of equations that has no solutions is called an inconsistent system. For example, consider the following linear system.

{y = 2 x - 1 y = 2 x + 2

To determine the number of solutions, the system can be graphed on a coordinate plane.

The lines do not intersect. The system has no solution and is therefore an inconsistent system.

Example

Finding the Solution and Classifying a System of Equations

Mark cannot go to Tearrik's party because he has recently started working with his father at a car dealership. At the dealership, Mark's father sells sedans and trucks.

Mark modeled the number of sedans and trucks sold with a system of equations.

{y = 2 x + 2 2 y - 4 x = 10 (I) (II)

Here,

x

represents the number of days since Mark started working in the agency. In Equation (I) the variable

y

represents the number of sedans sold. Similarly, in Equation (II)

y

represents the number of trucks sold. Mark wants to interpret this system of equations in terms of consistency and independence. Help him do this!

Hint

Graph both equations on the same coordinate plane.

Solution

To interpret the system in terms of consistency and independence, both linear equations will be drawn on the same coordinate plane. To do so, the slope-intercept form will be used. Since the first equation is already written in this form, the

y -

variable will be isolated in the second equation.

{y = 2 x + 2 2 y - 4 x = 10 (I) (II)

▼

(II):

Write in slope-intercept form

AddEqn

$(II):$ $LHS + 4 x = RHS + 4 x$

{y = 2 x + 2 2 y = 4 x + 10

DivEqn

$(II):$ $LHS / 2 = RHS / 2$

{y = 2 x + 2 y = \frac{4 x + 1 0}{2}

WriteSumFrac

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

{y = 2 x + 2 y = \frac{4 x}{2} + \frac{1 0}{2}

MovePartNumRight

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

{y = 2 x + 2 y = \frac{4}{2} x + \frac{1 0}{2}

CalcQuot

$(II):$ Calculate quotient

{y = 2 x + 2 y = 2 x + 5

Now that both equations are written in slope-intercept form, their slopes and

y -

intercepts can be used to draw the lines.

Next, recall the definitions of consistent, inconsistent, dependent, and independent systems.

Consistent System	A system of equations that has one or more solutions.
Independent System	A system of equations with exactly one solution.
Dependent System	A system of equations with infinitely many solutions.
Inconsistent System	A system of equations that has no solution.

With this information in mind, the lines will be considered one more time.

Since the lines do not intersect each other, the system has no solution. Therefore, it is an inconsistent system.

Pop Quiz

Classifying Systems

A system of equations can be consistent or inconsistent. In addition, a consistent system can be independent or dependent. Classify the following systems of equations in terms of consistency and independence.

Closure

Basketball and Soccer

With the content learned in this lesson, the challenge presented at the beginning can be solved. It has been said that the number of students in the US participating in high school basketball and soccer has steadily increased over the past few years.

The following table shows some information about this.

High School Sport	Number of Students Participating in $2015$ (Thousands)	Average Rate of Increase (Thousands per Year)
Basketball	$700$	$20$
Soccer	$400$	$40$

Consider the above table to answer the following questions.

a Write two equations in slope-intercept form to represent this situation.

b Graph the equations from Part A in the same coordinate plane. Use the graph to predict the approximate year when the number of students participating in these two sports will be the same.

Hint

a For both equations, let

x

be the number of years since

2015 .

b Pay close attention to the

x -

coordinate of the point of intersection of the lines.

Solution

a To start, the equation for the number of students participating in high school basketball will be written. The variables can be defined as follows.

Variable	Meaning of the Variable
$x$	Number of years since $2015$
$y$	Number of students that participate in high school basketball (thousands)

From the given table, it is known that in the year

2015,

700

thousands students participated in high school basketball. Also, the yearly average rate of increase is

20

thousand students. With this information, an equation can be written.

y = 20 x + 700

Next, the equation for the number of students participating in high school soccer will be written. The variables can be defined in a similar way as before.

Variable	Meaning of the Variable
$x$	Number of years since $2015$
$y$	Number of students that participate in high school soccer (thousands)

Again, it is seen in the table that in the year

2015,

400

thousands students participated in high school soccer. Also, the yearly average rate of increase is

40

thousand students. With this information, the second equation can be written.

y = 40 x + 400

b The equations found in Part A form a system of equations.

{y = 20 x + 700 y = 40 x + 400

Since both lines are already written in slope-intercept form, they can be graphed on the same coordinate plane.

The lines intersect at exactly one point. Therefore, this is a consistent and independent system. The solution is $x = 15$ and $y = 1000 .$ This means that $15$ years after $2015$ — the year $2030$ — there will be one million high school students participating in both sports.

Level 1

Level 2

Level 3

Solving Systems of Equations by Graphing

Exercises

The lines of a system of equations have been graphed.

What is the solution to this system?

The point of intersection of the two lines is the solution to the system of equations. Let's identify its coordinates.

The lines intersect at (3,5). This is the solution to the system. x=3 y=5

We can check the solution by substituting 3 and 5 for x and y, respectively, in both equations. If we obtain true statements, then our solution is correct. Conversely, if we obtain at least one false statement, then our solution is incorrect. Let's start by checking the solution in the equation y=x+2.

We got a true statement. Therefore, the solution satisfies the first equation. Let's now verify our solution in the equation y=2x-1.

We got a true statement again! This means that the solution satisfies both equations and therefore it is correct.

Jordan was solving the system of equations below when something strange happened.

⎩ ⎪ ⎨ ⎪ ⎧ y = 5 - 2 x 2 y + 4 x = 10

What can be seen if the lines are graphed on the same coordinate plane?

Let's draw the lines that represent the equations on the same coordinate plane. To do so, we need to rewrite them in slope-intercept form.

Now that the lines are written in slope-intercept form, we can use their slopes and y-intercept to graph them.

We see that the lines overlap each other. This means that the system has infinitely many solutions.

We actually do not need to graph the lines in order to know whether they overlap, intersect at only one point, or do not intersect at all. We only need to rewrite them in slope-intercept form.

If the lines have the same slope and the same y-intercept, then they overlap.
If the lines have the same slope but different y-intercepts, then they are parallel lines and therefore do not intersect.
If the lines have different slopes, then they intersect at exactly one point.

Interpret each system of linear equations in terms of consistency and independence.

⎩ ⎪ ⎨ ⎪ ⎧ y = 3 x - 4 y = 3 x + 6

⎩ ⎪ ⎨ ⎪ ⎧ y = - x + 9 y = 0.5 x

⎩ ⎪ ⎨ ⎪ ⎧ y = - 3 x - 8 y = 3 x + 8

We want to interpret the given system of equations in terms of consistency and independence. Let's first recall these definitions.

Consistent System	A system of equations that has one or more solutions.
Independent System	A system of equations with exactly one solution.
Dependent System	A system of equations with infinitely many solutions.
Inconsistent System	A system of equations that has no solution.

Let's now consider the given system. y=3x-4 y=3x+6 The equations are already written in slope-intercept form. We will use the slope and the y-intercept of each line to graph them on the same coordinate plane.

The lines are parallel and will never intersect each other. This means that the system of linear equations has no solution and therefore is an inconsistent system.

We again want to interpret the given system of equations in terms of consistency and independence. y=- x+9 y=0.5x Like before, the equations are already written in slope-intercept form. Let's use the slopes and y-intercepts to draw the lines!

Since the lines intersect exactly once, the system has exactly one solution. Considering the definitions we recalled in Part A, we can say that the system is a consistent and independent system.

We want to interpret the final system in terms of consistency and independence. y=-3x-8 y=3x+8 Like in the previous two parts, the lines are already written in slope-intercept form. This makes them easier to graph!

The lines intersect exactly once. Therefore, we know that the system has exactly one solution. This means that it is a consistent independent system.

Solve each system of equations graphically.

⎩ ⎪ ⎨ ⎪ ⎧ y = 3 x - 4 y = x + 4

⎩ ⎪ ⎨ ⎪ ⎧ y = - 3 x + 2 y = - 2 x + 3

⎩ ⎪ ⎨ ⎪ ⎧ 3 y = 3 x - 6 y + 0.5 x = - 5

To solve the system graphically, we should start by graphing the lines on the same coordinate plane. Luckily, both equations are already written in slope-intercept form. Let's use the slopes and y-intercepts to graph them!

The solution to the system is the point of intersection of the lines.

The lines intersect at (4,8). Therefore, the solution to the system is x=4, y=8.

Again, both lines are written in slope-intercept form. So, we will use their slopes and y-intercept to graph them.

The point where the graphs intersect is the solution to the system.

The lines intersect at (-1,5). Therefore, the solution to the system is x=- 1, y=5.

Before graphing the lines, we will rewrite them in slope-intercept form.

Now we are ready to graph the lines!

Let's identify the point of intersection between the lines.

Since the lines intersect at (-2,- 4), the solution to the system is x=- 2, y=- 4.

Consider the following system of equations.

⎩ ⎪ ⎨ ⎪ ⎧ y = x + 2 y = 5 - 2 x (I) (II)

Both equations have been graphed in the coordinate plane, along with four points

A,

B,

C

and

D .

Diagram showing the graphs of the equations and four points

Which point satisfies Equation (I)? Select all that apply.

Which point does not satisfy Equation (II)? Select all that apply.

Which point satisfies both equations? Select all that apply.

Let's first identify the graphs. To do so, we will identify the line that corresponds to Equation (I). The other line will correspond to Equation (II). We can see that Equation (I) is already written in slope-intercept form. Therefore, we can note its y-intercept just by looking at the equation. Slope-intercept Form:& y= mx+ b Equation (I):& y= 1x+ 2 The line that corresponds to Equation (I) intersects the y-axis at (0, 2). With this information, we can identify which line represents this equation. As we have already noted, the other line corresponds to Equation (II).

The points that satisfy Equation (I) lie on its line. From the coordinate plane, we see that B and C are both on this line, which means they satisfy the equation.

Now we need to find the points that do not satisfy Equation (II).

Points that do not fall on its line do not satisfy Equation (II). From the diagram, we see that A and C are not on this line. Therefore, neither A nor C satisfy Equation (II).

Points that are on both lines satisfy both equations. This means that the point of intersection of the lines, if there is one, satisfies both equations.

From the diagram, we see that B is the point of intersection of the lines. Therefore, B satisfies both equations.

Determine the solution to the system of equations shown in each graph.

Diagram with the graphs of two linear functions

The solution to a system of equations is the point where the lines intersect. We find the solution by finding the point of intersection and tracing it to the corresponding axes.

From the graph, we can see that this point is (3,-2). Therefore, the solution to the system is x=3, y=- 2.

We will again identify the point of intersection of the two lines in this system of equations, which is also the solution to the system.

The point of intersection of the lines is (2,1). This means that the solution to the system is x=2, y=1.

Solve each system of equations graphically.

⎩ ⎪ ⎨ ⎪ ⎧ 2 x + 4 y = 10 - x + 2 y = - 1

⎩ ⎪ ⎨ ⎪ ⎧ y - x = 5 y - 2 = 3 x

By graphing the given equations, we can determine the solution to the system. This will be the point at which the lines intersect, if there is any. To graph the lines, we will rewrite the equations in slope-intercept form.

We can now use the slopes and the y-intercepts of the lines to draw them on the same coordinate plane.

The solution to the system is the point of intersection of the lines.

Since the lines intersect at (3,1), the solution to the system is x=3, y=1.

Like in Part A, the very first thing we want to do is to rewrite the equations in slope-intercept form.

We can now use the slopes and y-intercepts to graph the lines.

The solution to the system is the point of intersection of the lines.

Since the lines intersect at (1.5,6.5), the solution to the system is x=1.5, y=6.5.

Solving Systems of Equations by Graphing

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