If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Digital SAT Math

Course: Digital SAT Math > Unit 2

Lesson 8: Graphs of linear systems and inequalities: foundations
Test prep
Digital SAT Math
Foundations: Algebra
Graphs of linear systems and inequalities: foundations
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

Graphs of linear systems and inequalities | Lesson

Google Classroom
A guide to graphs of linear systems and inequalities on the digital SAT

What are graphs of linear systems and inequalities problems?

Graphs of linear systems and inequality problems deal with graphs of the following in the x, y-plane:
In this lesson, we'll learn to:
  1. Identify solutions to linear systems and inequalities
  2. Match graphs with systems of linear equations and inequalities
This lesson builds upon an understanding of the following skills:
  • Solving linear equations and linear inequalities
  • Graphs of linear equations and functions
  • Solving systems of linear equations
You can learn anything. Let's do this!

How are systems of linear equations related to their graphs?

Systems of equations with graphing

Khan Academy video wrapper
Systems of equations with graphingSee video transcript

Intersection points as solutions

In a system of linear equations, each equation is represented by a line in the x, y-plane. The intersection of these lines represents the solution to the system.
For example, the graphs of y, equals, x and y, equals, minus, x intersect at left parenthesis, 0, comma, 0, right parenthesis, which is the solution to the system of equations.
Two lines are graphed in the xy plane. The one labeled y = x passes through the points (-2, -2), (0, 0), and (2, 2). The one labeled y= -x passes through the points (-2, 2), (0, 0), and (2, -2). The two lines intersect at the point (0, 0).

Try it!

Try: identify the solution of a system of equations by its graph
Two lines are graphed in the xy plane. One line passes through the points (-2, 5), (0, 1), and (2, -3). The other line passes through the points (-2, -4), (0, -2), and (2, 0).
In the graph above, the line with the positive slope has a slope of 1 and a y-intercept of
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
. The line with the negative slope has a slope of
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
and a y-intercept of 1.
The two lines intersect at left parenthesis
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
,
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
), which is the solution left parenthesis, x, comma, y, right parenthesis to the following system of equations y, equals, x, minus, 2 and y, equals, minus, 2, x, plus, 1.

How do I graph linear inequalities?

Intro to graphing two-variable inequalities

Khan Academy video wrapper
Intro to graphing two-variable inequalitiesSee video transcript

Representing solutions with shaded areas

When we learned to solve linear inequalities, we learned that multiple values can satisfy an inequality. How do we graph the infinite number of left parenthesis, x, comma, y, right parenthesis values that can satisfy a linear inequality?
We do it by shading the x, y-plane! A linear inequality is defined by a line in the x, y-plane. The line divides the plane into two halves, and which half of the plane we shade depends on the inequality sign.
It's the easier to determine which half of the plane to shade when the inequality is in slope-intercept form. When we have linear inequalities in slope-intercept form:
  • If y is greater than m, x, plus, b, shade above the line.
  • If y is less than m, x, plus, b, shade below the line.
But what about points on the line? In slope-intercept form, inequalities with...
  • Greater than (is greater than) or less than (is less than) signs do not include points on the line in the solution set. We use a dashed line to show that the points on the line are not included.
  • Greater than or equal to (is greater than or equal to) or less than or equal to (is less than or equal to) signs do include points on the line in the solution set. We use a solid line to show that points on the line are included.
For example, for start color #7854ab, y, is greater than, x, end color #7854ab, we draw y, equals, x as a dashed line and shade above the line:
The inequality y > x is graphed in the xy plane. The graph is a dashed line that passes through the points (-2, -2), (0, 0), and (2, 2). The area above the line is shaded.

Try it!

Try: verify solutions to a linear inequality
A dashed line in the xy plane passes through the points (-3, -1), (0, -2), and (3, -3). The line divides the plane into two halves. The half above the line is shaded.
The inequality y, is greater than, minus, start fraction, 1, divided by, 3, end fraction, x, minus, 2 is graphed in the x, y-plane above.
The point left parenthesis, 1, comma, 1, right parenthesis is
the shaded area, so it
a solution to the inequality.
The point left parenthesis, minus, 3, comma, minus, 4, right parenthesis is
the shaded area, so it
a solution to the inequality.
The point left parenthesis, 0, comma, minus, 2, right parenthesis is on the line y, equals, minus, start fraction, 1, divided by, 3, end fraction, x, minus, 2. It
a solution to the inequality because the is greater than sign indicates that points on the line
part of the solution set.

How do I graph systems of linear inequalities?

Intro to graphing systems of inequalities

Khan Academy video wrapper
Intro to graphing systems of inequalitiesSee video transcript

How do I identify the region representing a system of linear inequalities?

Two intersecting lines will always divide the x, y-plane into four regions. Points in each of the four regions represent solutions to a different system of linear inequalities.
In the graph below, the equations of the lines defining the four regions are:
y=3x−2y=−13x+43\begin{aligned} y &= 3x-2 \\\\ y &=-\dfrac{1}{3}x+\dfrac{4}{3} \end{aligned}
Replacing the equal signs with inequality signs lets us specify one of the four regions.
Two lines intersect in the xy-plane and divide the plane into four regions.
When we have linear inequalities in slope-intercept form:
  • If y is greater than m, x, plus, b, shade above the line.
  • If y is less than m, x, plus, b, shade below the line.
For example, let's look at the following system:
y≤3x−2y≥−13x+43\begin{aligned} y &\leq 3x-2 \\\\ y &\geq-\dfrac{1}{3}x+\dfrac{4}{3} \end{aligned}
This means the solutions to the system of linear inequalities are represented by the region below the line y, equals, 3, x, minus, 2 and above the line y, equals, start fraction, 1, divided by, 3, end fraction, x, plus, start fraction, 4, divided by, 3, end fraction:
The lines y=3x-2 and y=-1/3*x+4/3 intersect at (1, 1) and divide the xy-plane into four regions. The top-right region is shaded.

Try it!

TRY: DETERMINE THE INEQUALITY SIGNS FROM A GRAPH
The lines y=3x-2 and y=-1/3*x+4/3 intersect at (1, 1) and divide the xy-plane into four regions. The bottom-left region is shaded.
The the graph above represents a system of linear inequalities.
Because the region
the line y, equals, minus, start fraction, 1, divided by, 3, end fraction, x, plus, start fraction, 4, divided by, 3, end fraction is shaded, y is
or equal to minus, start fraction, 1, divided by, 3, end fraction, x, plus, start fraction, 4, divided by, 3, end fraction.
Because the region
the line y, equals, 3, x, minus, 2 is shaded, y is
or equal to 3, x, minus, 2.

Your turn!

practice: find the intersection of two lines
The graph of a line in the x, y-plane has a slope of 3 and passes through the origin. The graph of a second line has a slope of 6 and passes through the point left parenthesis, minus, 1, comma, minus, 9, right parenthesis. If the two lines intersect at the point left parenthesis, a, comma, b, right parenthesis, what is the value of a ?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

practice: match a linear inequality to its graph
A solid line in the xy plane passes through the points (-4, 0) and (0, -2). The line divides the plane into two halves. The top half is shaded.
If the shaded region in the graph above represents the solution set to an inequality, which of the following could be the inequality?
Choose 1 answer:

Practice: match a system of linear inequalities to its graph
x−y≥4y≤−34x−12\begin{aligned} &x-y \geq 4 \\\\ &y \leq -\dfrac{3}{4}x-\dfrac{1}{2} \end{aligned}
In which of the following does the shaded region represent the solution set in the x, y-plane to the system of inequalities above?
Choose 1 answer:

Things to remember

When a system of linear equations is graphed, the solution appears where the lines intersect.
When we have linear inequalities in slope-intercept form:
  • If y is greater than m, x, plus, b, shade above the line.
  • If y is less than m, x, plus, b, shade below the line.
In slope-intercept form, inequalities with...
  • Greater than (is greater than) or less than (is less than) signs do not include points on the line in the solution set. We use a dashed line to show that the points on the line are not included.
  • Greater than or equal to (is greater than or equal to) or less than or equal to (is less than or equal to) signs do include points on the line in the solution set. We use a solid line to show that points on the line are included.

Want to join the conversation?

Log in