Solving Systems of Linear Equations Algebraically and Graphically

In this lesson we will review what it means to solve a system of equations and then explore various algebraic and graphical methods for solving a system of equations. In the next lesson, we will apply our knowledge of solving systems of equations to real-world contexts.

LaTeX: x+2y=3

LaTeX: -2x+y=5

System of Linear Equations

$LaTeX: x^{2^{}}-y=3$

LaTeX: x=10+y

System of Equations Involving a Linear

Equation and a Quadratic Equation

To solve the system of equations means that we find the values of x and y that make both equations true or that satisfy both equations.

In the Introductory Section of this lesson, we will practice re-writing equations in a way that will help us solve systems of equations later in this lesson.

Re-Writing Equations
Problem #1
Problem #2

If we want to solve a system of linear equations, sometimes it is convenient to re-write each equation in slope-intercept form. It is also convenient to for one variable in terms of the other.

Slope-Intercept Form of a Line:

The slope-intercept form of a line is

LaTeX: y=mx+b

Where m is the slope of the line and b is the y-intercept of the line.

Here is an example:

Part A: Re-write the linear equation in slope-intercept form and give the slope and the y-intercept of the line.

LaTeX: x+2y=3

Solution:

Solve for y:

LaTeX: 2y=3-x

$LaTeX: y=\frac{3}{2}-\frac{1}{2}x$

$LaTeX: y=-\frac{1}{2}x+\frac{3}{2}$

Slope: $LaTeX: -\frac{1}{2}$

y-intercept: $LaTeX: \frac{3}{2}$

Part B: In the equation below, solve for x in terms of y:

LaTeX: x+2y=3

Solution:

LaTeX: x=3-2y

Problem #1

Part A: Re-Write the linear equation in slope-intercept form.

Solution:

Solve for y: LaTeX: 3x-y=1

LaTeX: -y=1-3x

LaTeX: y=-1+3x

LaTeX: y=3x-1

Part B: Give the slope of the line.

Solution:

Slope is 3

Part C: Give the y-intercept of the line.

Solution:

y-intercept is -1

Problem #2

Part A: Re-Write the linear equation in slope-intercept form.

Solution:

Solve for y: LaTeX: 10x=3-5y

LaTeX: 10x-3=-5y

$LaTeX: -2x+\frac{3}{5}=y$

Part B: Give the slope of the line.

Solution:

Slope is -2

Part C: Give the y-intercept of the line.

Solution:

y-intercept is $LaTeX: \frac{3}{5}$

Part D: In the equation, solve for x in terms of y:

Solution:

Solve for x in terms of y: LaTeX: 10x=3-5y

$LaTeX: x=\frac{3}{10}-\frac{5}{10}y$

LaTeX: 10x=3-5y

Solving Systems of Equations Algebraically

In this next contextual example, we will see a real-world scenario where a system of equations comes into play. Later in this module, we will solve this system of equations.

Contextual Example: New Mexico Artist’s Cutting Boards
Three Methods for Solving Systems of Linear Equations
Real World Model: Domain

A woodworker in New Mexico makes beautiful art from various types of wood and other natural materials. Below is an example of one of the cutting boards available at Sandia Mountain Woodworks. This particular board is made from 3 types of wood: Walnut, Mahogany and Maple.

Screenshot 2024-09-04 at 11.31.46 PM.png

In this problem, we will explore an example connected to creating cutting boards from only two types of wood: Mahogany and Walnut.

Cutting Board Example

Suppose the artist makes 2 different kinds of cutting boards. Type 1 is made from Walnut and Mahogany. This type of cutting board uses 4 strips of Walnut and 2 strips of Mahogany.

Another type of cutting board, Type 2, also is made from Walnut and Mahogany, but it requires 2 strips of Walnut and 3 strips of Mahogany.

Suppose that the strips of each type of wood used in these two types of cutting boards is the same length and width.

If the total cost of wood material to make Type 1 is approximately $35 and the total cost of wood material to make Type 2 is approximately $30, how much does each strip of each of the types of wood cost?

You will return to this example in the next lesson.

In this lesson, we will explain three methods for solving systems of linear equations:

1. Elimination

1. Solving Systems of Linear Equations by the Method of Elimination

We will explore a system of two linear equations. When we solve a system of linear equations using the method of elimination, we eliminate on of the variables and create one equation with only one variable.

Watch the Youtube Video until 4:44 to see examples of how to solve a system of linear equations using the Method of Elimination.

Example:

The example below shows the method of elimination in a step-by-step fashion

	System of Equations
	Multiply the first equation by 3 so that its y term has the same coefficient as the y-term in the 2nd equation except it will be negative
$LaTeX: \left(9x-3y\right)+\left(2x+3y\right)=21+1$	Add the 2nd equation from the first
	Simplify
	Solve for x
$LaTeX: 2\left(2\right)+3y=1$	Substitute the value of x into either one of the first two equations
	Solve for y
$LaTeX: \left(2,-1\right)$	Write Solution as an Ordered Pair or a Point

Elimination Problems:

Problem 1:

Solve the system of equations using the method of Elimination. Show your steps and give your solution as an ordered pair.

LaTeX: 2x-y=4

LaTeX: 4x-3y=12

Solution:

Multiply the 1st equation by 2:

$LaTeX: 2\left(2x-y\right)=2\left(4\right)$

Simplify:

LaTeX: 4x-2y=8

Subtract the 2nd equation from this new equation:

LaTeX: 4x-2y=8

LaTeX: -4x+3y=-12

LaTeX: y=-4

Substitute this value of y into the 1st equation:

$LaTeX: 2x-\left(-4\right)=4$

And solve for x:

LaTeX: 2x+4=4

LaTeX: 2x=0

LaTeX: x=0

Solution is $LaTeX: \left(0,-4\right)$

2. Substitution

2. Solving Systems of Linear Equations by the Substitution

When we solve a system of equations using the method of substitution, we use one equation and solve for one variable in terms of the other and then substitute that expression into the other equation.

Watch this Youtube video to see some examples of how to solve a system of linear equations using the method of substitution.

Example:

In the table below, we show a step-by-step procedure for solving the system of linear equations by the Method of Substitution

	System of Equations
	From the first equation, we can express y in terms of x:
$LaTeX: 2x+3\left(3x-7\right)=1$	Substitute this expression for y into the second equation
$LaTeX: x=\frac{22}{11}$	Simplify and solve for x
$LaTeX: 3\left(2\right)-y=7$	Substitute the value of x into the expression for y:
$LaTeX: \left(2,-1\right)$	Write solution as an ordered pair

Substitution Problem:

Problem 1:

Solve the system of equations using the method of Substitution. Show your steps.

LaTeX: x-7y=-11

LaTeX: 5x+2y=-18

Solution:

Solve the first equation for x in terms of y:

$LaTeX: x-7y=-11\Rightarrow x=7y-11$

Substitute the expression for x into the second equation:

LaTeX: 5x+2y=-18

$LaTeX: 5\left(7y-11\right)+2y=-18$

Solve for y:

$LaTeX: 5\left(7y-11\right)+2y=-18$

LaTeX: 35y-55+2y=-18

$LaTeX: 37y=37\rightarrow y=1$

Finally substitute the value of y into the expression for x:

LaTeX: x=7y-11

$LaTeX: x=7\left(1\right)-11=-4$

Write your solution as an ordered pair

$LaTeX: \left(-4,1\right)$

3. Graphing

3. Solving Systems of Linear Equations by Graphing

When we solve a system of linear equation using graphing, we graph each line and identify the intersection point of the two lines.

Example:

Solve the system of linear equations using graphing:

LaTeX: 7x-8y=-12

LaTeX: -4x+2y=3

Using Desmos, we graphed the two equations and then click on the intersection point to see that the solution to the system is the point (0,1.5)

Screenshot 2024-09-05 at 12.49.29 AM.png

Graphing Problem:

Problem 1:

Use the Geogebra Applet below to explore the graphical solutions of these systems of equations. Write your answers as ordered pairs.

LaTeX: 6x-5y=8

LaTeX: -12x+2y=0

Solution: $LaTeX: \left(0.33,-2\right)$

Screenshot 2024-09-05 at 12.56.44 AM.png

Problem 2:

Use the Geogebra Applet again to explore the graphical solutions of these systems of equations. Write your answers as ordered pairs.

LaTeX: 2x+3y=20

LaTeX: 7x+2y=53

Solution: $LaTeX: \left(7,2\right)$

Screenshot 2024-09-05 at 12.58.54 AM.png

When does a system of linear equations have no solution?

Note: There are some systems of linear equations that have no solution. This can happen when the two lines are parallel and therefore have no intersection point.

Example of a System of Linear Equations that has no solution

Consider the system of linear equations below:

LaTeX: 2x-3y=4

LaTeX: 4x-6y=12

Suppose we try to solve this system of equations by elimination.

We would multiple the first equation by 2

$LaTeX: 2\left(2x-3y\right)=2\left(4\right)$

LaTeX: 4x-6y=12

Simplify and subtract the 2nd equation from the first:

LaTeX: 4x-6y=8

LaTeX: -4x+6y=-12

LaTeX: 0=-4

We know that this is impossible, so we have no solution to the system.

Now let’s graph the two equations using Desmos. The graphs as shown below. We can see that the two lines are parallel and therefore never intersect.

Screenshot 2024-09-04 at 11.44.10 PM.png

When does a system of linear equations have an infinite number of solutions?

Note: There are some systems of linear equations that have an infinite number of solutions.

This can happen when the two lines are the same line expressed in different ways and therefore have an infinite number of intersection points.

Example of a System of Linear Equations that has an infinite number of solutions

Consider the system of equation show below:

LaTeX: 5x-y=4

LaTeX: 10x-2y=8

If we try to solve this system using substitution, in the 1st equation, we can solve for y in terms of x as follows:

$LaTeX: 10x-2\left(5x-4\right)=8$

And Simplify:

LaTeX: 10x-10x+8=8

LaTeX: 8=8

Since this is a true statement, the solution to the systems is the set of all real numbers that lie on the line

LaTeX: 5x-y=4

We may also notice that the 2nd equation is a multiple of the 1st equation. If we multiply the first equation by 2, we get the second equation. This means that the two equations are the same line.

Solving Systems of Linear Equations Algebraically and Graphically

In this lesson we will review what it means to solve a system of equations and then explore various algebraic and graphical methods for solving a system of equations. In the next lesson, we will apply our knowledge of solving systems of equations to real-world contexts.

Re-Writing Equations

Problem #1

Problem #2

Contextual Example: New Mexico Artist’s Cutting Boards

Three Methods for Solving Systems of Linear Equations

Real World Model: Domain

When you have completed the above lesson, click the next button in Canvas to progress.