All of these equations are linear, and can be expressed in slope-intercept form; y=mx+b. m is the slope, and b is the y-intercept.
tip: when replacing variables with values, always add parenthesis ( )
tip: f(x) can be rewritten as y
5) the slope is -2/3, y-intercept is 5. y=(-2/3)x+(5) -> f(x)=(-2/3)x+5
6) the slope is 7/4, y-intercept is 0. y=(7/4)x+(0) -> f(x)=(7/4)x
When you are tasked with finding slope and y-intercept using points, there are a few methods. The first is to plug in the known values.
7) The y-intercept is where x=0 and where the line passes through the y- axis. The point to do this is (0,-3). Using this we know that b=-3.
Now that we have the y-intercept, all we need is the slope (m).
We will use the formula m=(y2-y1)/(x2-x1)
m=(-3-0)/(0-2)=(-3/-2)=(3/2)
m=(3/2)
y=(3/2)x-3 -> f(x)=(3/2)x-3
8) There is no point given where x is equal to 0, so we will use a slightly different method. First we will find slope using m=(y2-y1)/(x2-x1).
m=((-2)-(-6))/(5-10)=4/-5=-4/5
m=-4/5
Next, to find the y-intercept (b), we will plug in the slope and one point to the equation.
(-2)=(-4/5)(5)+b
-2=-4+b
b=2
y=(-4/5)x+(2) -> f(x)=(-4/5)x+(2)
9) The y-intercept of this equation is (0,-1), which is where the line crosses the y-axis, therefore b=-1. Next, for slope, we can simply count the grid from a point using rise/run. From the y-intercept, we can count up 2, right 1, which is equal to 2/1, or 2.
y=(2)x+(-1) -> y=2x-1 -> f(x)=2x+1
10) The y-intercept of this equation is (0,3), so b=3. Counting the grid with rise/run we get down 1, right 2, which is -1/2.
y=(-1/2)x+(3) -> f(x)=(-1/2)x+(3)
11) Javier is starting with $140, which will be his y-intercept, b. b=140. He earns $25 per week, 25/1 or 25. m=25.
a) y=25x+140 -> f(x)=25x+140
b) Slope is $25/week, y-intercept is $140.
c) To find the solution to this equation, where he needs $410, we need to plug that value of $410 into the equation as y so we know how long it will take.
(410)=(25)(x)+(140)
x=10.8
It will take Javier 10.8 weeks to save up for the new bike.