Linear Equations

MATH 102Contemporary Math

S. Rook

Overview

• Section 7.1 in the textbook:– Solving linear equations– Graphing linear equations using intercepts– Slope– Slope-intercept form of a line

Solving Linear Equations

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

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

Solving Linear Equations (Example)

Ex 1: Solve:

a) 8 – 2y = 4 + 3y

b) 4x – 7 = 7x + 14

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numbers have been replaced by variables!

7

Solving Linear Equations (Example)

Ex 2: Solve each for x:

a)

b) 2x + 3y = 6

x

z

Graphing Linear Equations Using Intercepts

1010

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

1111

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

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

Graphing Linear Equations Using Intercepts (Example)

Ex 3: Graph by first finding the intercepts:

a) 3x + 2y = 12

b) 4x – 3y = 16

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:

12

12

xx

yy

x

ym

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 slopeb is the y-coordinate of the y-intercept (0, b)

• To utilize the slope-intercept form of a line, y MUST be ISOLATED

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

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?

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)