Alg III Linear Equations
Basic Understandings
The standard form for a linear equation is AX

BY = C.
Whenever an equation has X or Y or both to the first power then it is a linear equation.
Another form for a linear equation is Y = mX

b, where m is the slope and b is the
y – intercept.(known as the slope intercept equation of a line)
The slope of a line can also be determined from looking at a graph and finding the 𝑟𝑖𝑠𝑒 𝑟𝑢𝑛
or
∆ 𝑦
∆ 𝑥
The slope of a line can also be found given two points. The equation is
( Y
2
– Y
1
)/(X
2
– X
1
) or
∆ 𝑦
∆ 𝑥
A line that slants up and to the right has a positive slope.
A line that slants up and to the left has a negative slope.
Parallel lines have the same slope.
Perpendicular lines have negative reciprocal slopes.
A linear equation that looks like X = # is a vertical line and has no slope.
A linear equation that looks like Y = # is a horizontal line and has a slope of zero.
The y-intercept is found when X=0 or (0,y).
The x-intercept is found when Y= 0 or (x,0).
Graphing a linear equation
If you have an equation in standard form you can use a table, use the intercept method, or change the equation to slope intercept form and use the slope and intercept to graph.
Example 2x – 3y = 6
Table method pick values for x and then find y and plot the points and draw the line.
X Y
3 0
6 2
-3 -4
Intercept method. Put in zero for x and find y. Put in zero for y and find x.
( 0, -2) ( 3, 0) Now plot the points and draw the line.

Slope intercept method is to change the equation to slope intercept form.
2x – 3y = 6 -3y = -2x + 6 y = 2/3 x – 2
Now you place a point on the y-axis at -2 and the go up 2 and right 3 and place another point and draw your line.
On the grid below plot all three methods.
Finding the slope
If you have an equation in slope intercept form y = mx + b it is the number in front of x.(exceptions are vertical and horizontal lines mentioned above) ex y = 8x – 12 the slope is 8
If you have an equation in standard form them change it to slope intercept form. Ex 2x + 5y = 10
5y = -2x + 10 y = -2/5 x + 2 the slope is -2/5
If you have two points then use (Y
2
– Y
1
)/(X
2
– X
1
) or ∆𝑦 / ∆𝑥 ex (1,4) and (5,6) (6-4)/(5-1) or 2/4 or ½
If you have a graph then just determine the rise/run from the graph. What is the slope of both lines on the grid below.
One is -1/2 because it is rising by one and running left by 2
The other is 3/1 because it is rising by three and running right by 1

Writing the Equation of a line
If you are given the slope and the y-intercept, then use the slope intercept form of a line.
Ex slope is -3 and intercept (0,5) y - -3x + 5
Ex m = ¾ and b = -2 Y = ¾ x – 2
If you are given the slope and a point, then you can use the point slope equation of a line (Y – Y
1
) = m(X – X
1
) or you can use y = mx + b.
Example 1 slope is 4 and point is (3,-2)
Plug the point into equation and get y by itself Plug point into equation and solve for b
(Y - -2) = 4(X – 3) -2 = 4(3) + b
Y + 2 = 4X – 12 -2 = 12 + b
Y = 4X – 14 -14 = b
Y = 4X – 14
Example 2 slope is ¾ and point is (-2,4)
Plug the point into equation and get y by itself Plug point into equation and solve for b
(Y – 4) = ¾(X - - 2) 4 = ¾(-2) + b
Y – 4 = ¾ X + 6/4 4 = -6/4 + b
Y = ¾ X + 4 + 6/4 4 + 6/4 = b
Y = ¾ X + 5.5 5.5 = b
Y = ¾ X + 5.5
If you are given two points, then you find the slope of the line use one of the points and one of the two methods above to find the equation.
Example 3 points (4,2) and (2,6) slope is (6-2)/2-4) or 4/-2 or -2
(Y – 2) = -2(X – 4) 2 = -2(4) + b
Y – 2 = -2X + 8 2 = -8 + b
Y = -2X + 10 10 = b
Y = -2X + 10
Example 4 points (-4,2) and (8,5) slope is (5-2)/(8 - - 4) or 3/12 or ¼
(Y – 5) = ¼(X – 8) 5 = ¼(8) + b
Y – 5 = ¼ X – 2 5 = 2 + b
Y = ¼ X + 3 3 = b
Y = ¼ X + 3

Horizontal and Veritical Lines
X = 4 is a vertical line passing through the X – intercept of 4
Y = -3 is a horizontal line passing through the Y – intercept of – 3
Working with the knowledge of horizontal and vertical lines.
Find the equation of a vertical line that contains the point (-2 , 7)?
A vertical lines has an equation of X = _____. Since my line must contain the point
( -2 , 7) and the X value of the point is – 2 my equation is X = -2 .
Find the equation of a horizontal line that contains the point (5 , 11)?
A horizontal line has an equation of Y = _____. Since my line must contain the point
(5 , 11) and the Y value of the point is 11 my equation is Y = 11 .
Find the equation of a line that contains the point (3 , -4) and is parallel to the line whose equation is Y = 2 ?
The line Y = 2 is a horizontal line. If my line is parallel it must also be a horizontal line.
A horizontal line has an equation Y = _______. Since my line must contain the point
(3 , -4) and the Y value of my point is – 4 my equation is Y = - 4.
Find the equation of a line that contain the point (7 , 8) and is perpendicular to the line whose equation is Y = -4
?
The line Y = - 4 is a horizontal line. If my line is perpendicular it must be a vertical line.
A vertical line has an equation of X = ____. Since my line must contain the point (7 , 8) and the X value of my point is 7 my equation is X = 7 .

Draw the following equations on the graph below: a) Y = 6 b) X = -3 c) 2X + Y = -6
Problems d) X – 5Y = 10 e) Y = 2X -3 f) Y = ½ X + 2
Name the slope and Y – intercept of each of the following equations:
1) X = 7 m = b = 2) Y = -3 m =
3) X = -2
5) Y = 2X – 5
7) 2X – 3Y = 4 m = m = m = b = 4) Y = 12 b = 6) Y = 4/5 X b = 8) X + 5Y = 8 m = m = m = b = b = b = b =
Find the equation of a line given the following information:
9) It has no slope and contains the point (4 , 3) __________________________
10) It is a horizontal line and contains the point (-2 , 6) _________________________
11) It contains the point (-3 , 7) and is perpendicular to the line Y = 8 ________________
12) It has a slope of zero and contains the point (3, -8) ___________________
13) It contains the point (4 , -2) and is parallel to the line Y = 6 ___________________
14) It contains the points (3 , 4) and (3 , -10) __________________________
15) It contains the points (4 , -2) and ( -5 , -2) _________________________
16) It contains the point ( 0 , 6) and has a slope of -4 _________________________
17) It contains the point ( 2 , 7) and it’s slope is undefined ____________________
18) It contains the points ( 4 , 8) and ( 5 , 2) _____________________
19) It contains the point ( 3 , -4) and is perpendicular to the line X = 2 ______________
20) It is parallel to the X-axis and contains the point (-2 , 4) ______________________
Tell whether the following lines are parallel, perpendicular, or intersecting
21) y = 2x + 6 y = - ½ x + 8 22) y = 5x – 7 y = 5x + 9
23) 3y + x = 7 y – 3x = 4 24) 2x + 5y = 10 4x + 10y = -8
25) 2x – 3y = 7 2x + 3y = 15 26) 4x + 3y = 8 3x – 4y = -19
27) 8x – y = 10 y = - 1/8 x + 2 28) 2x + y = 8 6x + 3y = 4