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Math 10: Slopes and Lines

Q: What is the equation of a line?

A: The equation of a line is y=mx+b, where

m = slope

b = y-intercept

Q: What is slope?

A: The slope of a line is a measure of how steep the line is. A larger slope means that the line is steeper, while a smaller slope corresponds to a flatter line.

The equation for slope is:

Slope Equation
So essentially, slope is just the change in y divided by the change in x.

Slope Graphic

Let’s look at a graph with three different lines. Note that the green line has the largest slope, while the blue line has the smallest slope.

Slopes of three lines

Q: What is the y-intercept?

A: In the equation of a line, y=mx+b, the y-intercept is b. It is the y-value where the line crosses the y-axis.

Using the same three lines, let’s find their y-intercepts.

y-intercepts of three lines

Lastly, let’s look at the same three lines with their equations in y=mx+b form.

Equations of three lines

Any line is defined by just two pieces of information: the slope, m, and the y-intercept, b.

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Q: The equation of a line is y=2x -7. What is the y-intercept?

A: It is easy enough to simply compare the equation to the standard form, y=mx+b, and see that b = -7.

However, the general way to find the y-intercept is to set x=0, and solve for y.

y = 2(0) – 7
y = 0 – 7
y = -7

Either way you do it, the y-intercept is -7. This is the location where the line crosses the y-axis.

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Q: Can you show me how to find the slope of a line with the equation (y+3)=5(x-2)?

A: First, turn the equation into slope-intercept form, y=mx+b. Do this by isolating for y in the equation.

y + 3 = 5x – 10
y = 5x – 10 – 3
y = 5x – 13

Compare this to the general equation for a line: y=mx+b.

You should see that m = 5 and b = -13. This means that the slope is 5.

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Q: Can you show me the steps on how to work this out? What is the y-intercept of a line with the equation (y-3)=5(x+2)? Thanks.

A: Again, turn the equation into slope-intercept form, y=mx+b, by isolating for y:

y – 3 = 5x + 10
y = 5x + 10 + 3
y = 5x + 13

Compare this to the general equation for a line: y=mx+b.

You should see that b = 13. The y-intercept is 13. This means the line crosses the y-axis at y=13.

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Q: What is the slope of a line that contains the two points (0,3) and (4,5) ?

A: Recall that the equation for slope is:



You have two points, (0,3) and (4,5), so you have (x1,y1) and (x2,y2). Put them into the formula:

m = (5-3) / (4-0)
m = 2 / 4
m = 1/2

So the slope is 1/2. This means it goes up 1, right 2.

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Q: Please show me how to write an equation that is the slope-intercept form of the equation of the line that passes through (1,2) and is parallel to 4x-2y=6

A: This looks hard at first, but it’s actually not too bad.

First, let’s rearrange the 4x-2y=6 line so that it’s in slope-intercept form, y=mx+b

4x-2y=6
4x-6=2y

Divide both sides by 2:

4x/2 -6/2 = y
2x – 3 = y
y=2x -3

So the slope of this line is 2, because m=2.

If a line is parallel to it, it must have the exact same slope. Parallel lines always have identical slopes. We know that this second line passes through the point (1,2), which means that x=1 and y=2. We already know that m=2, so put it all into the equation of a line and solve for b:

y=mx+b
2=2(1)+b
2=2+b
2-2=b
b=0

So the y-intercept is zero.

The equation of the line is:

y=2x

To see a graph of this line on Wolfram Alpha, click here.

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