Unit B: Linear Equations

B.1 Introduction to Linear Equations and Slope

What is a linear equation?

• An equation that has no operation other than addition, subtraction, and multiplication of a variable by a constant.

• The variables may not be multiplied together, or appear in a denominator.

• Do not contain variables with an exponent other than 1.

Examples and non-examples

Examples• 5x-3y=7

• X=9

• 6s=-3t-15

• Y=1/2 x

Non-Examples• 7a+4b2 = -8

• X+xy=1

• Y=1/x

Example 1

• State whether each is a linear equation. Explain.

• Y=10-5x• Y=x4-5• H=2xy

What is a linear equation?

• The SOLUTIONS to a linear equation are the values of (x,y) that make the equation balance.

• To find solutions to any equation, plug in whatever you want to x, then solve for y.

• How many solutions can you find to the equation y=2x?

• The solutions can be represented by using the order pairs, or you can draw them on a graph.

• Let’s draw the solutions to y=2x on the graph.

Intercepts• Intercepts are the points on a graph where

the line passes through the x- and y-axis. • What would the x- and y-intercepts of this

graph be?

Slope

• The slope of a line is the ratio of the change in the y-coordinates to the change in the x-coordinates.

• Basically, it represents how steep the line is.

• There are two ways we can find slope.

Rise over Run

• When given a graph, it is pretty easy to find the slope of the line.

• 1. Find two points where the x and y are whole numbers.

• 2. Count how many up or down you have to go, then count how many left or right you have to go to get from one point to another.

Example 2

• Find the slope of this line.

• Remember: Rise OVER run

Example 3

• Find the slope of this line.

• Remember: Rise OVER run

• Pos/neg zero undefined

Finding Slope Using Ordered Pairs

• Use the formula:

• m= y-y/x-x

• Where the first y and x must come from the same ordered pair.

Example 4

• Find the slope of the line that passes through the points (-3,2) and (1, -4).

• Find the slope of the line that passes through the points (-4,-3) and (2,1).

Equations of Lines

• The equation of a line in slope-intercept form is

• Y=mx+b where m is the slope and b is the y-intercept.

• Example:What is the slope and the y-intercept of Y=2x+4?

Example 5

• Write the equation of a line that has a slope of 3 and a y-intercept of -1.

Example 6

• What is the slope and y intercept of the line 2x+4 = 2y?

• What is the slope and y-intercept of the line 12x-4y=8x?

Example 7

• Write the equation of the line that goes through (3,1) and (2,2).

• Write the equation of the line that goes through (-1,-1) and (4,5).

Graphing Lines Given the Equation

• To graph a line, put it in slope-intercept form (solve for y).

• Then, graph the y-intercept.• Lastly, use the slope to graph another point.

Example 8

• Graph the line x-y=6.

Example 9

• Write the equation of the line shown below.

Equations of Vertical and Horizontal Lines

• Horizontal: Remember HOY– H=horizontal– 0=zero slope=– Y=# is the equation

• Vertical: Remember VUX– V=vertical– U=undefined slope– X=# is equation

Example 10

• Graph the line that has an undefined slope and passes through (1,2).

• Graph the horizontal line that passes through (0,1).