1. MATH 102 Contemporary Math S. Rook
Linear Equations
MATH 102
Contemporary Math
S. Rook

2. Overview Section 7.1 in the textbook: Solving linear equations
Graphing linear equations using intercepts
Slope
Slope-intercept form of a line

3. Solving Linear Equations

4. Linear Equations
Standard Form of a Linear Equation: Ax + By = C where A, B, and C are real numbers
All variables in a linear equation are raised to the first power
In standard form when all variables are on one side and constants on the other
Think of a linear equation as a scale
Like a scale, an equation has two sides separated by the =
The scale (equation) MUST remain balanced at all times
What you do to one side, you MUST do to the other

5. Solving Linear Equations
Recall that to solve a linear equation, we must isolate the variable by
Ensuring that the variable appears on only one side of the equation
Adding or subtracting the same number to both sides
Multiplying or dividing the same number on both sides
e.g. Solve 5x + 40 = 3x – 10 for x

6. Solving Linear Equations (Example)
Ex 1: Solve: a) 8 – 2y = 4 + 3y b) 4x – 7 = 7x + 14

7. Solving a Linear Equation with More than One Variable
Now, consider solving ax – b = c for x
It is the same process as solving 2x – 3 = 7 for x!
Do NOT be intimidated by the fact that the numbers have been replaced by variables!

8. Solving Linear Equations (Example)
Ex 2: Solve each for x: a) b) 2x + 3y = 6

9. Graphing Linear Equations Using Intercepts

10. Cartesian Plane
The Cartesian Plane consists of two number lines – one horizontal and one vertical
The point of intersection of the two lines is known as the origin and has coordinates (0, 0)
A coordinate (x, y) consists of two numbers:
x represents the horizontal (left to right) direction from the origin
y represents the vertical (top to bottom) from the origin 10

11. Cartesian Plane (Continued)
Split into four sections called quadrants based on the sign of x and y of the coordinate (x, y)
A point that lies on the axes is known as quadrantal 11

12. x and y-intercepts
x-intercept: where a graph of a linear equation crosses the x-axis
Written in coordinate form as (x, 0)
y-intercept: where a graph of a linear equation crosses the y-axis
Written in coordinate form as (0, y)

13. Graphing By Intercepts
Specialized form of using a table of values
Find the x-intercept
Find the y-intercept
Plot the intercepts on the graph and draw the line
Two points make a line

14. Graphing Linear Equations Using Intercepts (Example)
Ex 3: Graph by first finding the intercepts: a) 3x + 2y = 12 b) 4x – 3y = 16

15. Slope-Intercept Form of a Line

16. Definition and Properties of Slope
Slope (m): the ratio of the change in y (Δ y) and the change in x (Δ x)
Quantifies (puts a numerical value on) the “steepness” of a line
Given 2 points on a line, we can find its slope:

17. Slope (Example)
Ex 4: Find the slope between the points: a) (2, 5) and (6, 8) b) (9, 1) and (6, 4)

18. Slope-Intercept Form
Slope-intercept form: a linear equation in the form y = mx + b where
m is the slope
b is the y-coordinate of the y-intercept (0, b)
To utilize the slope-intercept form of a line, y MUST be ISOLATED

19. Slope-Intercept Form of a Line (Example)
Ex 5: State the slope and y-intercept of the line: a) y = 4x – 3 b) 2x + y = 7

20. Slope-Intercept Form of a Line (Example)
Ex 6: A health club charges a yearly membership fee of $95, and members must pay $2.50 per hour to use its facilities.
a) What are the fixed and variable costs?
b) Let y represent total membership cost and x represent the number of hours using the club’s facilities. Write an equation.
c) How much would it cost if Ivan used the club’s facilities for 100 hours?
d) How many hours did Jillian use the club facilities last year if her bill was $515?

21. Summary
After studying these slides, you should know how to do the following:
Solve a linear equation
Graph a linear equation using its intercepts
Find the slope of a pair of points or a linear equation
Additional Practice:
See problems in Section 7.1
Next Lesson:
Modeling with Linear Equations (Section 7.2)