Solving systems of equations by graphing

Here, you will learn how to solve a system of equations by graphing, including linear and nonlinear systems. You will also learn how to interpret graphs to identify solutions.

Students first learn how to solve systems of equations in the 8 th grade when they learn linear systems. They expand that knowledge as they move through high school math classes.

What is solving systems of equations by graphing?

Solving systems of equations by graphing is the process of solving two or more algebraic equations (linear or nonlinear) that share the same variables by sketching their graphs and observing possible points of intersection.

The point or points of intersection of two or more equations on the coordinate graph are the solution(s) to the system. This is because the point(s) of intersection of the two functions is where the two functions (equations) are equal.

When graphing systems of equations that have points of intersection, they are said to be consistent systems with common points.

Systems can exist with no solutions or infinite solutions.

What is solving systems of equations by graphing?

Common Core State Standards

How does this apply to 8 th grade math and high school math?

• Grade 8 – Expressions and Equations (8.EE.C.8a)
  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

• High School Algebra – Reasoning with Equations and Inequalities: (HSA-REI.C.6)
  Solve systems of linear equations exactly and approximately (example, with graphs), focusing on pairs of linear equations in two variables.

• High School Algebra – Reasoning with Equations and Inequalities: (HSA-REI.C.7)
  Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y=- \, 3x and the circle x^2+y^2=3.

How to solve a system by graphing

In order to solve a system by graphing, you need to:

1. Graph both equations on the same set of axes.
2. Locate any points of intersection on the graph.
3. State the solution(s).

Solving systems of equations by graphing examples

Example 1: solving linear system

Solve the following system by graphing:

1. Graph both equations on the same set of axes.

Both equations are in slope-intercept form, so they are ready to graph.

For y=2x+1\text{:}

• The y -intercept b=1 , so plot the point (0, \, 1).
• The slope m=2=\cfrac{2}{1} , so from the y -intercept move 2 units up, and 1 unit right to (1, \, 3).
• Draw a straight line through these two points across the plotting area.

For y=4x+3\text{:}

• The y -intercept b=3 , so plot the point (0, \, 3).
• The slope m=4=\cfrac{4}{1} , so from the y -intercept, move 4 units up and 1 unit right to (1, \, 7).
• Draw a straight line through these two points across the plotting area.

2Locate any points of intersection on the graph.

The two graphs intersect at the point (- \, 1, \, 1).

3State the solution(s).

The solution is x=- \, 1 and y=- \, 1.

You can check your solution by substituting x=- \, 1 into each equation and finding y=- \, 1 (or vice versa):

• y=2x+1=2\times(- \, 1)+1=- \, 2+1=- \, 1
• y=4x+3=4\times(- \, 1)+3=- \, 4+3=- \, 1

Example 2: solving nonlinear system

Solve the nonlinear system by graphing.

Graph the equations (linear and/or nonlinear) on the same set of axes.

Calculate a handful of points on the curve y=x^{2}+1 by substituting values of x into the equation to find the corresponding values of y.

Plot the points and connect using a single smooth curve.

The line y=1 is a horizontal line intersecting the y -axis at (0, \,1) as the values of x are independent to y. For any x -value, y=1 always.

Locate any points of intersection on the graph.

The two graphs intersect at the point (0, \, 1).

Example 3: solving linear system with no solution

Solve this linear system by graphing.

Graph the equations (linear and/or nonlinear) on the same set of axes.

You can use several strategies to graph the linear equations.

Let’s use the slope intercept form to graph, rearranging each equation to the form y=mx+b. This is the same as solving each equation for y.

For 4x+2y=- \, 8\text{:}

• The y -intercept b=- \, 4 , so plot the point (0, \, - \, 4).
• The slope m=- \, 2=\cfrac{- \, 2}{1} , so from the y -intercept move 2 units down and 1 unit right to (1, \, - \, 6).
• Draw a straight line through these two points across the plotting area.
  
  

For 6x+3y=12\text{:}

• The y -intercept b=4 , so plot the point (0, \, 4).
• The slope m=- \, 2=\cfrac{- \, 2}{1} , so from the y -intercept move 2 units down and 1 unit right to (1, \, 2).
• Draw a straight line through these two points across the plotting area.
  
  

Example 4: solving nonlinear system with no solution

Solve the nonlinear system by graphing.

Graph the equations (linear and/or nonlinear) on the same set of axes.

Calculate a handful of points on the curve y=x^{2}+1 by substituting values of x into the equation to find the corresponding values of y.

Plot the points and connect using a single smooth curve.

The line y=5 is a horizontal line intersecting the y -axis at (0, \, 5) as the values of x are independent to y. For any x -value, y=5 always.

Example 5: solving nonlinear system with a cubic

Solve the nonlinear system of equations.

Graph the equations (linear and/or nonlinear) on the same set of axes.

Calculate a handful of points on the curve y=x^{3} by substituting values of x into the equation to find the corresponding values of y.

Plot the points and connect using a single smooth curve.

For y=x\text{:}

• The y -intercept b=0 , so plot the point (0, \, 0).
• The slope m=1=\cfrac{1}{1} , so from the y -intercept move 1 unit up and 1 unit right to (1, \, 1).
• Draw a straight line through these two points across the plotting area.
  
  

Locate any points of intersection on the graph.

The two graphs intersect at the three points: (- \, 1, \, - \, 1), \, (0, \, 0) and (1, \, 1).

Example 6: solving nonlinear systems with circle (estimate of solutions)

Find approximate solutions to the nonlinear system by graphing.

Graph the equations (linear and/or nonlinear) on the same set of axes.

The equation x^2+y^2=4 represents a circle that has its center at (0, \, 0) and its radius is \sqrt{4}=2. The points where the circle intersect each axis are found by substituting x=0 and y=0 into the equation of the circle.

When x=0,

\begin{aligned}0^{2}+y^{2}&=4 \\\\ y^{2}&=4 \\\\ y&=\pm \, \sqrt{4}= \pm \, 2 \end{aligned}

So when x=0, \, y=2 and y=- \, 2 giving the two coordinates (0, \, 2) and (0, \, - \, 2).

When y=0,

\begin{aligned}x^{2}+0^{2}&=4 \\\\ x^{2}&=4 \\\\ x&=\pm \, \sqrt{4}= \pm \, 2 \end{aligned}

So when y=0, \, x=2 and x=- \, 2 giving the two coordinates (2, \, 0) and (- \, 2, \, 0).

Plot all four points onto the set of axes and connect them with a circle centered at (0, \, 0).

The line x=1.2 is a vertical line intersecting the x -axis at (1.2, \, 0) as the values of y are independent to x. For any y -value, x=1.2 always.

Teaching tips for solving systems of equations by graphing

• Have students work in collaborative groups using poster paper type graph paper to sketch systems of equations and identify solutions. Have students present their sketches to the class or create a gallery walk.

• Plot graphs using online tools.

• Allow students to use a calculator, emphasizing the use of brackets when substituting a negative number raised to a power.

Easy mistakes to make

• Sketching graphs incorrectly
  Although there are several strategies to sketch the various functions, the easiest way to graph them if you do not recall other strategies is to:
  
  • Complete a table of values for each of the equations.
  • Try to solve the equations for y (especially for linear functions).
  • Select values for x and substitute them into the equations to find the values of y.
  • Sketch the points on the coordinate plane and connect the points to form the equations.
  • Look for the points of intersection.

• Thinking that a system of linear equations that are not parallel does not have a solution
  When sketching lines (or any function), be sure to extend them as long as possible in order to find the point of intersection. Sometimes, if lines are not extended, it might look like they do not intersect.
  For example, these lines are not parallel, which means they do intersect. Be sure to draw the lines long enough so that you can see the point of intersection.
  
  
  As you can see, extending the lines helps to see the point of intersection.
  
  

• Thinking there is only \bf{1} point of intersection
  Remember linear and quadratic functions can intersect at 0, \, 1 or 2 points. If the lines have multiple points of intersection, make sure they match the correct value of x and y. Lines and circles can intersect at 0, \, 1, or 2 points too.

• Thinking that the most you can have is \bf{2} points of intersection
  A system of equations that are circles or circles and parabolas, or any combination of conic sections can produce as many as 4 points of intersection.
  For example, in this case of a system that is comprised of a parabola and circle, there are 4 points of intersection.
  
  

Related systems of equations lessons

• Systems of equations
• Intersecting lines
• Nonlinear system of equations

The next lessons are

• Number patterns
• What is a function
• Laws of exponents

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