GED Mathematical Reasoning: Slope And Equations

Slope-intercept form

Slope-Intercept form looks like:

y = mx + b

Where the number in the m spot, the coefficient of x, is the slope of the line and the number in the b spot, tacked onto the end, is the y-coordinate of the y-intercept of the graph. The y-intercept is where the graph crosses the y-axis.

In slope-intercept form, the x and y remain as variables. They represent all the different ordered pair points on the line.

Let’s look at a visual so that you can see the relationship between an equation in slope-intercept form and its graph.

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If we identify two points on the line that go through the cross-hairs of the grid, we see that the “rise over run” is 4 over 3. And since the line is going downhill from left to right, the slope of the line is negative three-fourths. Do you see how this slope measure corresponds to the coefficient of x in the equation?

Also notice that the graph crosses the y-axis at the point (0, 4). The value of b in the equation is equal to 4, which is the y-coordinate of this ordered pair.

coord17

 

Point-slope form 

Point-slope form looks like:

y - y_1 = m(x - x_1)

Where “m” is the slope and (x_1, y_1) represents the x and y coordinates of any point on the line.

As in slope-intercept form, the x and y remain as variables. They represent all the different ordered pair points on the line.

 

Example 1

Use Slope-Intercept form to find the equation of the line that passes through the point (2, -4) and has a slope of m = -3.

 

Let’s start by writing the slope-intercept form: y = mx + b

In this case, we know the slope but not the y-intercept. The point we’re given is on the line, but it doesn’t correspond to a point on the y-axis. So our strategy will be to use the slope-intercept form as a formula. We’ll substitute what we know and then solve for what we don’t know.

We can substitute m with the given slope of -3. The point we’re given is not the y-intercept but we know that the point (2, -4) is on the line so we can substitute it for x and y. If we let x = 2 and y = -4, we can then solve the equation for b, the y-coordinate of the y-intercept.

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So the value for b, the y-coordinate of the y-intercept, is positive 2. Now that we know the value for the slope AND y-intercept of the line, we can state the equation in slope-intercept form. It is:

y = -3x + 2

Where the coefficient of “x”, -3, is the slope and positive 2 is the y-coordinate of where the graph crosses the y-axis. As an ordered pair, we would write the y-intercept as: (0, 2)

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Example 2

Use Point-Slope form to find the equation that passes through the points (-7, -5)  and (-3, -2). State your answer in slope-intercept form.

 

Let’s start by writing the Point-slope form: y - y_1 = m(x - x_1)

For point slope-form, we need the slope and a point. But in this example, we’re given two points. Our strategy will be to use the two given points along with the slope formula to find the slope of the line. From there, we’ll use the slope and either point to determine the point-slope form of the equation. Our final step will be to transform the point-slope form of the equation into slope-intercept form.

To find the slope of the line passing through the points: (-7, -5) and (-3, -2) we’ll use the slope formula:

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

Let’s use (-7, -5) for the first point and (-3, -2) for the second point.

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m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(-2) - (-5)}{(-3) - (-7)} = \frac{(-2) + 5}{(-3) + 7} = \frac{3}{4}

This gives us a slope of: \frac{3}{4}. Let’s substitute this for m in the point-slope form:

y - y_1 = m(x - x_1) \rightarrow  y - y_1 = \frac{3}{4}(x - x_1)

Now we can use either point for (x_1, y_1) or (x_2, y_2)(since they both lie on the line). I’ll use the first point.

So the equation of the line in point-slope form is:

y - (-5) = \frac{3}{4}(x - (-7))

y + 5 = \frac{3}{4}(x + 7)

Since the instructions of this example ask us to write the equation in slope-intercept form, we aren’t quite finished yet. We must transform this point-slope form equation into slope-intercept form. We can do so by solving the equation for y.

y + 5 = \frac{3}{4}(x + 7)

y + 5 = \frac{3}{4}x + \frac{21}{4}

y + 5 - 5 = \frac{3}{4}x + \frac{21}{4} - 5

y = \frac{3}{4}x + \frac{21}{4} - 5

y = \frac{3}{4}x + \frac{1}{4}

So the final equation in slope-intercept form is: y = \frac{3}{4}x + \frac{1}{4}

 

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