Lesson 1b – Linear Equations
MAT12x – Intermediate Algebra
MINI LESSON
Lesson 1b – Linear Equations
Lesson Objectives:
1. Identify LINEAR EQUATIONS
2. Determine slope and y-intercept for a LINEAR EQUATION
3. Determine the x-intercept for a LINEAR EQUATION
4. Draw graphs of LINEAR EQUATIONS
5. Graph Horizontal and Vertical lines
6. Write LINEAR EQUATIONS
The topic of linear equations should be at least slightly familiar to students starting Intermediate Algebra.
The basics are covered here with reminders of important ideas and concepts that will be heavily utilized in
the next lesson.
What is a Linear Equation?
A LINEAR EQUATION is an equation that can be written in the form:
y = mx + b
with slope, m, and y-intercept (0, b)
This is the SLOPE-INTERCEPT form for the equation of a line.
Slope
The slope of a line is denoted by the letter m. Given any two points, (x1, y1), (x2, y2), on a line, the slope is
determined by computing the following ratio:
y !y
m= 2 1
x2 ! x1
Note: Slope is also referred to as the “change in y” over the “change in x” and is a measure of steepness
and direction for a given line.
Problem 1
WORKED EXAMPLE – DETERMINE SLOPE OF A LINEAR EQUATION
Find the slope of the line through the points (2, -5) and (-3, 4).
m=
4 ! (!5) 4 + 5 9
9
=
=
=!
!3! (2) !5 !5
5
Note: The slope is negative indicating the line decreases from left to right. If slope is positive, then the line
increases from left to right.
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Lesson 1b – Linear Equations
MAT12x – Intermediate Algebra
Y-intercept
Also called the VERTICAL INTERCEPT, this is the special ordered pair with coordinates (0, b). 0 is the value
of x (input) and the resulting output (b) is the y-coordinate of the y-intercept. The y-intercept is often used
to help when graphing a linear equation and/or to determine the initial output value in an application
setting.
Problem 2
WORKED EXAMPLE – DETERMINE Y-INTERCEPT FOR A LINEAR EQUATION
Find the y-intercept (also called the vertical intercept) for the equation y = -2x + 6.
Method 1: Read the value of b from y=mx + b form.
This equation is written in the form y = mx + b. Therefore, the y-intercept is (0, 6).
If the equation is NOT in y = mx + b form, isolate y in the equation to make it so.
Method 2: Solve for y when x = 0
Replace x with 0 to get y = -2(0) + 6 = 0 + 6 = 6. The ordered pair (0,6) is the y-intercept.
Problem 3
MEDIA EXAMPLE – DETERMINE SLOPE/Y-INTERCEPT FOR LINEAR EQUATN
Complete the table below. Note: Method 1 will be used to find the y-intercept for all the problems below
because we are also asked to determine the slope. Practice finding the y-intercept using Method 2 as well.
Equation
y = mx + b form
Slope
Y-intercept
a) y = -2x +5
b) y = 2 – x
c) y = 3 x + 2
4
d) 2x – y = 4
e) 3x + 2y = 6
f) 8x = 4y
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Lesson 1b – Linear Equations
MAT12x – Intermediate Algebra
X-intercept
Also called the HORIZONTAL INTERCEPT, this is the special ordered pair with coordinates (a, 0). 0 is the
value of y (output) and the resulting input (a) is the x-coordinate of the x-intercept. The x-intercept is often
used to help when graphing a linear equation and/or to determine the final input value in an application.
Problem 4
WORKED EXAMPLE – FIND X-INTERCEPT FOR A LINEAR EQUATION
Find the x-intercept (also called the horizontal intercept) for the equation y = -2x + 6.
Method: Replace the value of y with 0 then solve for the value of x.
0 = -2x + 6
-6 = -2x
-6/-2 = x
3=x
Problem 5
The x-intercept is (3, 0).
MEDIA EXAMPLE – FIND X-INTERCEPT FOR A LINEAR EQUATION
For each of the following problems, determine the x-intercept as an ordered pair using the methods above.
Equation
Show Work
a) y = -2x +5
b) y = 2 – x
c) y = 3 x + 2
4
d) 2x – y = 4
e) 3x + 2y = 6
f) 8x = 4y
3
Lesson 1b – Linear Equations
Problem 6
MAT12x – Intermediate Algebra
YOU TRY – DRAW GRAPHS OF LINEAR EQUATIONS
3
Use the equation y = ! x + 9 for all parts of this problem. Label all plotted points.
2
a) Use the x-intercept and y-intercept to draw the graph of the line. Show your work to find these points.
PLOT and LABEL the intercepts on the graph then connect them to draw your line.
x-intercept : ( ___ , ___ )
y-intercept: ( ___ , ___ )
Work to find x-intercept
Work to find y-intercept
b) Determine the coordinates of two OTHER ordered pairs and use those to graph the line. PLOT and
LABEL the points you use.
Remember from your previous math work that you can create a t-table with ordered pairs other than the
intercepts OR you can insert the equation into your Y= list on your TI 83/84 calculator then use the TABLE
feature to identify two other ordered pairs. Remember to enter as y1=(-3/2)x + 9.
Ordered pair 1: ( ___ , ___ )
Ordered pair 2: ( ___ , ___ )
NOTICE that your graphs for parts a) and b) should look exactly the same.
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Lesson 1b – Linear Equations
MAT12x – Intermediate Algebra
Special Linear Equations
The following media problem will introduce you to two special types of linear equations: horizontal and
vertical lines.
Problem 7
MEDIA EXAMPLE – GRAPHING HORIZONTAL/VERTICAL LINES
a) Use the grid below to graph the equation y = -2. Identify the slope and y-intercept.
b) Use the grid below to graph the equations x = 5. Identify the slope and y-intercept.
Equations of Vertical Lines
• equation: x = a
• x-intercept: (a, 0)
• y-intercept: none
• slope: m is undefined
Equations of Horizontal Lines
• equation: y = b
• x-intercept: none
• y-intercept: (0, b)
• slope: m = 0
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Lesson 1b – Linear Equations
MAT12x – Intermediate Algebra
Writing Equations of Lines
Critical to a thorough understanding of linear equations is the ability to write the equation of a line given
different pieces of information. The following process will work for almost every situation you are presented
with and will be illustrated several times in the media problems to follow.
Step 1: Determine the value of the slope, m.
Step 2: Determine the coordinates of one ordered pair.
Step 3: Plug the values for the ordered pair, and the value for the slope, into y = mx + b
Step 4: Solve for b
Step 5: Use the values for m and b to write the resulting equation in y = mx + b form.
Problem 8
MEDIA EXAMPLE – WRITING EQUATIONS OF LINES
For each of the following, find the equation of the line that meets the following criteria:
a) Slope m = -4 passing through the point (0, 3).
b) Passing through the points (0, -2) and (1, 5)
c) Passing through the points (-2, -3) and (4, -9)
d) Parallel to y = 3x – 7 and passing through (2, -5)
e) Passing through (2, 4) with an x-intercept of -2.
f) Vertical line passing through (-3, 5).
g) Horizontal line passing through (-3, 5).
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Lesson 1b – Linear Equations
Problem 9
MAT12x – Intermediate Algebra
YOU TRY – WRITING LINEAR EQUATIONS FROM GRAPHS
Use the given graph to help answer the questions below. Assume the line intersects grid corners at integer
(not decimal) values.
a) Is the line above increasing, decreasing, or constant?
Read the graph from left to right.
b) What is the vertical (y) intercept?
________________
y-intercept? ______________
Also, plot, label on the graph.
c) What is the horizontal (x) intercept?
x-intercept? _______________
Also, plot, label on the graph.
d) What is the slope (m)?
Show work at right to compute.
Hint: Use two points from the graph.
Work to find slope:
Slope = ___________
e) What is the equation of the line in y=mx + b form?
Hint: Use the slope and y-intercept from above.
Equation of the line:
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Lesson 1b – Linear Equations
Problem 10
MAT12x – Intermediate Algebra
YOU TRY – WRITING EQUATIONS OF LINES
a) Find the equation of the line passing through the points (1,4) and (3,-2) and write your equation in the
form y = mx + b. Show complete work in this space.
b) What is the vertical (y) intercept for this equation? Show work or explain your result.
c) What is the horizontal (x) intercept for this equation? Show complete work to find this.
Problem 11
YOU TRY – HORIZONAL AND VERTICAL LINES
a) Given the ordered pair (2, -3)
• Write the equation of the vertical line through this point. _________
• Identify the slope of the line: __________
• What is the y-intercept? ____________
• What is the x-intercept? ____________
b) Given the ordered pair (2, -3)
• Write the equation of the horizontal line through this point. _________
• Identify the slope of the line: __________
• What is the y-intercept? ____________
• What is the x-intercept? ____________
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Lesson 1b – Linear Equations
Problem 12
MAT12x – Intermediate Algebra
WORKED EXAMPLE – WRITING EQUATIONS OF LINES
Write an equation of the line to satisfy each set of conditions.
a) Line contains the points (-3, 5) and (0, 1)
Slope
y-intercept
Use the ordered pairs (-3, 5)
Given y-intercept in the ordered
and (0, 1) to compute slope.
pair (0, 1) that was provided
then b = 1.
Equation
Plug m and b into y = mx + b
y="
m = 1" 5 = "4
0 " ("3)
4
3
x +1
3
b) Line contains points (-4, -3) and (2, 6)
!
Slope
y-intercept
Equation
!
Use the ordered pairs (-4, -3)
Not given y-intercept. Pick one
Plug m and b into y = mx + b
and (2, 6) to compute slope.
given ordered pair and plug m
3
and ordered pair into y = mx+b.
y= x+3
Solve
for
b.
6
!
(!3)
6
+
3
9
3
2
m=
2 ! (!4)
=
2+4
=
6
=
2
Using (2, 6) then 6 =
3
2
(2) + b
so 6 = 3 + b or b = 3.
!
c) Line has the following graph:
!
Slope
Identify two ordered pairs from
the graph and use them to
determine the slope.
(5, 0) and (3, -1)
y-intercept
Read the y-intercept from the
graph. Ordered pair is (0, -3).
Therefore b = -3.
Equation
Plug m and b into y = mx + b
y=
1
2
x-3
m = !1! (0) = !1 = 1
3! (5)
!2
2
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Lesson 1b – Linear Equations
MAT12x – Intermediate Algebra
ANSWERS – YOU TRY PROBLEMS
You Try Problem 6:
x-intercept (6, 0), y-intercept (0, 9), Answers vary on other ordered pairs, graphs for parts a) b) should be
the same
You Try Problem 9:
a) decreasing b) (0, 4) c) (4, 0) d) slope = -1 e) y = -x + 4
You Try Problem 10:
a) y = -3x + 7
b) (0,7)
c) (7/3, 0) or (2.33, 0)
You Try Problem 11:
a) equation x = 2, slope is undefined, no y-intercept, x-intercept (2, 0)
b) equation y = -3, slope m = 0, y-intercept (0, -3), no x-intercept
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