7.4 The slope-intercept form of a line

Find the slope of the line passing through $(2,1)$ and $(6,3)$ . Graph this line on the graph of problem 2 below.

Find the slope of the line passing through $(3,4)$ and $(5,5)$ . Graph this line.

Compare the lines of the following problems. Do the lines appear to cross? What is it called when lines do not meet (parallel or intersecting)? Compare their slopes. Make a statement about the condition of these lines and their slopes.

The equation $y=mx+b$ is called the slope-intercept form of the equation of a line. The number $m$ is the slope of the line and the point $(0,b)$ is the $y\text{-intercept}$ .

The slope, $\mathrm{m,}$ of a line is defined as the steepness of the line, and it is the number of units that $y$ changes when $x$ changes 1 unit.

The formula for finding the slope of a line through any two given points $(x_1,y_1)$ and $(x_2,y_2)$ is

$m=\frac{y_2-y_1}{x_2-x_1}$

The fraction $\frac{y_2-y_1}{x_2-x_1}$ represents the $\frac{\text{Change}\text{in}y}{\text{Change}\text{in}x}.$

As we look at a graph from left to right, lines with positive slope rise and lines with negative slope decline.

Parallel lines have the same slope.

Horizontal lines have 0 slope.

Vertical lines have undefined slope (or no slope).

$y=3x+4$

$y=2x+9$

$y=9x+1$

$y=7x+10$

$y=-4x+5$

$y=-2x+8$

$y=-6x-1$

$y=-x-6$

$y=-x+2$

$2y=4x+8$

$4y=16x+20$

$-5y=15x+55$

$-3y=12x-27$

$y=\frac{3}{5}x-8$

$y=\frac{2}{7}x-12$

$y=\frac{-1}{8}x+\frac{2}{3}$

$y=\frac{-4}{5}x-\frac{4}{7}$

$-3y=5x+8$

$-10y=-12x+1$

$-y=x+1$

$-y=-x+3$

$3x-y=7$

$5x+3y=6$

$-6x-7y=-12$

$-x+4y=-1$

$(1,6),(4,9)$

$(1,3),(4,7)$

$(3,5),(4,7)$

$(6,1),(2,8)$

$(0,5),(2,-6)$

$(-2,1),(0,5)$

$(3,-9),(5,1)$

$(4,-6),(-2,1)$

$(-5,4),(-1,0)$

$(-3,2),(-4,6)$

$(9,12),(6,0)$

$(0,0),(6,6)$

$(-2,-6),(-4,-1)$

$(-1,-7),(-2,-9)$

$(-6,-6),(-5,-4)$

$(-1,0),(-2,-2)$

$(-4,-2),(0,0)$

$(2,3),(10,3)$

$(4,-2),(4,7)$

$(8,-1),(8,3)$

$(4,2),(6,2)$

$(5,-6),(9,-6)$

Do lines with a positive slope rise or decline as we look left to right?

Do lines with a negative slope rise or decline as we look left to right?

Make a statement about the slopes of parallel lines.

$3.8x+12.1y=4.26$

$8.09x+5.57y=-1.42$

$10.813x-17.0y=-45.99$

$-6.003x-92.388y=0.008$

$(5.56,9.37),(2.16,4.90)$

$(33.1,8.9),(42.7,-1.06)$

$(155.89,227.61),(157.04,227.61)$

$(0.00426,-0.00404),(-0.00191,-0.00404)$

$(88.81,-23.19),(88.81,-26.87)$

$(-0.0000567,-0.0000567),(-0.00765,0.00764)$

( [link] ) Simplify ${(x^2{y}^3{w}^4)}^0$ .

( [link] ) Solve the equation $3x-4(2-x)-3(x-2)+4=0$ .

( [link] ) When four times a number is divided by five, and that result is decreased by eight, the result is zero. What is the original number?

( [link] ) Solve $-3y+10=x+2$ if $x=-4$ .

( [link] ) Graph the linear equation $x+y=3$ .