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SAT

Course: SAT > Unit 6

Lesson 3: Heart of Algebra: lessons by skill
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Heart of Algebra: lessons by skill
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Graphing linear equations | Lesson

Google Classroom

What are "Graphing linear equations" questions, and how frequently do they appear on the test?

Graphing linear equations questions deal with linear equations and their graphs in the x, y-plane. For example, the graph of y, equals, 2, x, minus, 1 is shown below.
In this lesson, we'll learn to:
  1. Identify features of linear graphs from their equations
  2. Write linear equations based on graphical features
  3. Determine the equations of parallel and perpendicular lines
  4. Identify solutions to systems of linear inequalities as regions in the x, y-plane
On your official SAT, you'll likely see 3 to 4 questions that test your ability to graph linear equations.
You can learn anything. Let's do this!

What are the features of lines in the x, y-plane?

Intro to slope

Khan Academy video wrapper
Intro to slopeSee video transcript

Features of lines in the x, y-plane

The slope

The slope of a line describes its direction and steepness.
  • A line that trends upward from left to right has a positive slope.
  • A line that trends downward from left to right has a negative slope.
  • The steeper the line is, the larger the
    of its slope is.
The slope is equal to the ratio of a line's change in y-value to its change in x-value. We can calculate the slope using any two points on the line, left parenthesis, x, start subscript, 1, end subscript, comma, y, start subscript, 1, end subscript, right parenthesis and left parenthesis, x, start subscript, 2, end subscript, comma, y, start subscript, 2, end subscript, right parenthesis:
start text, s, l, o, p, e, end text, equals, start fraction, start text, c, h, a, n, g, e, space, i, n, space, end text, y, divided by, start text, c, h, a, n, g, e, space, i, n, space, end text, x, end fraction, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction
Example: Line ell contains the points left parenthesis, minus, 1, comma, 2, right parenthesis and left parenthesis, 4, comma, 12, right parenthesis. What is the slope of line ell ?
A horizontal line has a slope of 0 since all points on the line have the same y-coordinate (so the change in y is 0).
A vertical line has an undefined slope since all points on the line have the same x-coordinate (so the change in x is 0).

The y-intercept

The y-intercept of the line is the point where the line crosses the y-axis. This point always has an x-coordinate of 0. All non-vertical lines have exactly one y-intercept.

The x-intercept

The x-intercept of the line is the point where the line crosses the x-axis. This point always has a y-coordinate of 0. All non-horizontal lines have exactly one x-intercept.

Try it!

Try: identify the features of a graph
The graph of line m is shown above.
The y-intercept of the line is located at left parenthesis, 0, comma
  • 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
right parenthesis.
The x-intercept of the line is located 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
comma, 0, right parenthesis.
We can use the intercepts to calculate the slope of the line. The slope of the line is
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
  • an exact decimal, like 0, point, 75
.

How do I tell the features of lines from linear equations?

Converting to slope-intercept form

Khan Academy video wrapper
Converting to slope-intercept formSee video transcript

How do I interpret an equation in slope-intercept form?

Lines in the x, y-plane are visual representations of linear equations. The slope-intercept form of a linear equation, y, equals, m, x, plus, b, tells us both the slope and the y-intercept of the line:
  • The slope is equal to m.
  • The y-intercept is equal to b.
For example, the graph of y, equals, start color #7854ab, 3, end color #7854ab, x, start color #ca337c, minus, 7, end color #ca337c has a slope of start color #7854ab, 3, end color #7854ab and a y-intercept of start color #ca337c, minus, 7, end color #ca337c.
Because the slope-intercept form shows us the features of the line outright, it's useful to rewrite any linear equation representing a line in slope-intercept form.
Example: What is the slope of the graph of 3, x, plus, 4, y, equals, 12 ?

Try it!

Try: identify slope and intercept from a linear equation
2, x, plus, y, equals, 3
To rewrite the above equation in slope-intercept form, we can isolate y by
both sides of the equation.
When 2, x, plus, y, equals, 3 is graphed in the x, y-plane:
  • The slope of the line is
    • 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 y-intercept of the line is
    • 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
    .

How do I write linear equations based on slopes and points?

Slope-intercept equation from two points

Khan Academy video wrapper
Slope-intercept equation from two pointsSee video transcript

What information do I need to write a linear equation?

We can write the equation of a line as long as we know either of the following:
  • The slope of the line and a point on the line
  • Two points on the line
In both cases, we'll be using the information provided to find the missing values in y, equals, m, x, plus, b.

The slope and a point

When we're given the slope and a point, we have values for x, y, and m in the equation y, equals, m, x, plus, b, and we just need to plug in the values and solve for the y-intercept b.
Example: If line a has a slope of 2 and passes through the point left parenthesis, 1, comma, 3, right parenthesis, what is the equation of line a ?
Note: if the given point is the y-intercept, then we just need to plug in the slope for m and the y-intercept for b. No calculation needed!

Two points

When we're given two points, we must first calculate the slope using the two points, then plug in the values of x, y, and m into y, equals, m, x, plus, b to find b.
Example: Line b passes through the points left parenthesis, minus, 2, comma, 4, right parenthesis and left parenthesis, 1, comma, minus, 5, right parenthesis. What is the equation of line b ?

Try it!

TRY: WRITE the equation of a line
The line shown above passes through the points left parenthesis, 0, comma
  • 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
right parenthesis and left parenthesis, 2, comma, 2, right parenthesis.
The slope of the line is
  • 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 equation of the line in slope-intercept form is:
y, equals

How do I write equations of parallel and perpendicular lines?

Parallel & perpendicular lines from graph

Khan Academy video wrapper
Parallel & perpendicular lines from graphSee video transcript

What are the features of parallel and perpendicular lines?

In the x, y-plane, lines with different slopes will intersect exactly once.
Parallel lines in the x, y-plane have the same slope. Parallel lines do not intersect unless they also completely overlap (i.e., are the same line).
Perpendicular lines in the x, y-plane have slopes that are
of each other. Perpendicular lines form 90, degrees angles.
The graph below shows lines ell, m, and n.
  • Line start color #7854ab, ell, end color #7854ab has a slope of 2.
  • Line start color #ca337c, m, end color #ca337c also has a slope of 2. It is parallel to line start color #7854ab, ell, end color #7854ab.
  • Line start color #208170, n, end color #208170 has a slope of minus, start fraction, 1, divided by, 2, end fraction. It is perpendicular to both lines start color #7854ab, ell, end color #7854ab and start color #ca337c, m, end color #ca337c.
This means we can write the equation of a parallel or perpendicular line based on a slope relationship and a point on the line.

Let's look at some examples!

Lines p and q are graphed in the x, y-plane. Line p is represented by the equation y, equals, minus, 2, x, minus, 5. If line q is parallel to line p and passes through the point left parenthesis, 0, comma, 4, right parenthesis, what is the equation of line q ?
Line ell is represented by the equation y, equals, minus, 3, x, plus, 2. What is the equation of a line that is perpendicular to line ell and intersects line ell at left parenthesis, minus, 3, comma, 11, right parenthesis ?

Try it!

TRY: identify the features of parallel and perpendicular lines
The line shown in the graph above has a slope of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
and a y-intercept of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
.
The slope of a line parallel to the line above must have a slope of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
.
The slope of a line perpendicular to the line above must have a slope of
  • Your answer should be
  • an integer, like 6
  • a proper fraction, like 1, slash, 2 or 6, slash, 10
  • an improper fraction, like 10, slash, 7 or 14, slash, 8
  • an exact decimal, like 0, point, 75
.
The slope-intercept form equation of a perpendicular line that intersects the line shown above at left parenthesis, 0, comma, 3, right parenthesis is:
y, equals

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 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.
When we have linear inequalities in slope-intercept form:
  • If y is greater than the 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:

Try it!

TRY: DETERMINE THE INEQUALITY SIGNS FROM A GRAPH
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: match an equation and its graph
Which of the following is the graph of the equation y, equals, minus, start fraction, 1, divided by, 2, end fraction, x, plus, 3 in the x, y-plane?
Choose 1 answer:

Practice: write an equation based two points
Line ell in the x, y-plane passes through the points left parenthesis, minus, 1, comma, minus, 2, right parenthesis and left parenthesis, 2, comma, 7, right parenthesis. Which of the following equations describes line ell ?
Choose 1 answer:

Practice: find a point on a perpendicular line
In the x, y-plane, line ell has a y-intercept of minus, 7 and is perpendicular to the line with equation y, equals, minus, start fraction, 1, divided by, 4, end fraction, x. If the point left parenthesis, 5, comma, c, right parenthesis is on line ell, what is the value of c ?
  • 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 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

start text, s, l, o, p, e, end text, equals, start fraction, start text, c, h, a, n, g, e, space, i, n, space, end text, y, divided by, start text, c, h, a, n, g, e, space, i, n, space, end text, x, end fraction, equals, start fraction, y, start subscript, 2, end subscript, minus, y, start subscript, 1, end subscript, divided by, x, start subscript, 2, end subscript, minus, x, start subscript, 1, end subscript, end fraction
The slope-intercept form of a linear equation, y, equals, m, x, plus, b, tells us both the slope and the y-intercept of the line:
  • The slope is equal to m.
  • The y-intercept is equal to b.
We can write the equation of a line as long as we know either of the following:
  • The slope of the line and a point on the line
  • Two points on the line
Parallel lines in the x, y-plane have the same slope.
Perpendicular lines in the x, y-plane have slopes that are negative reciprocals of each other.
When we have linear inequalities in slope-intercept form:
  • If y is greater than the m, x, plus, b, shade above the line.
  • If y is less than m, x, plus, b, shade below the line.

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