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+4b 2 = -8 X+xy=1 Y=1/x

Example 1 State whether each is a linear equation. Explain. Y=10-5x Y=x 4 -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).