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# Slope-intercept from two points

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Complete the equation of the line through (-6,-5)(-4,-4)

Jul 28, 2017

#1
+2298
+1

To figure out the equation of a line that passes through the given points (-6, -5) and (-4, -4), you must first know the standard form of a line. It is the following:

$$y=mx+b$$

Let m = slope of the line

Let b = the y-intercept (the point where the line touches the y-axis)

The first step is to figure out the slope of the line. How do we do that, you may ask? All you do is remember the slope formula.

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

We already have enough information to calculate the slope, m. We do this by substituting the given points into the formula.

$$m=\frac{-5-(-4)}{-6-(-4)}$$Simplify the fraction into simplest terms by evaluating the numerator and denominator separately.

$$m=\frac{-5+4}{-6+4}$$Of course, subtracting a negative is the same as adding a positive.

$$m=\frac{-1}{-2}$$The negatives in the numerator and denominator cancel each other out.

$$m=\frac{1}{2}$$

Great! We know the slope! Now, the only variable to figure out next is b, the y-intercept. We can do this by plugging in points of points on the line in the equation.

$$y=\frac{1}{2}x+b$$

In other words, to solve for b, you must plug in a point we know is one the line (either (-6,-5) or (-4,-4)) for x and y. I'll choose (-4,-4):

$$y=\frac{1}{2}x+b$$Plug in the coordinate (-4,-4) in its appropriate spots and then solve for b.

$$-4=\frac{1}{2}*-4+b$$Now, solve for b.

$$-4=-2+b$$Add 2 on both sides.

$$-2=b$$

Now that we have solved for both m and b, the equation that passes through the points (-6,-5) and (-4,-4) is $$y=\frac{1}{2}x-2$$.

Do you need your answer in point-slope form? No problem! Remember the point-slope form

$$y-y_1=m(x-x_1)$$

Of course, m is the slope again. We have already calculated that. Let's substitute that in.

$$y-y_1=\frac{1}{2}(x-x_1)$$

$$y_1\hspace{1mm}\text{and}\hspace{1mm}x_1$$ represent a point on the line. You can either substitute the first or the second set of coordinates. It doesn't matter. However, in the end, your answer should be one of these:

$$y+5=\frac{1}{2}(x+6)$$

$$y+4=\frac{1}{2}(x+4)$$

.
Jul 28, 2017

#1
+2298
+1

To figure out the equation of a line that passes through the given points (-6, -5) and (-4, -4), you must first know the standard form of a line. It is the following:

$$y=mx+b$$

Let m = slope of the line

Let b = the y-intercept (the point where the line touches the y-axis)

The first step is to figure out the slope of the line. How do we do that, you may ask? All you do is remember the slope formula.

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

We already have enough information to calculate the slope, m. We do this by substituting the given points into the formula.

$$m=\frac{-5-(-4)}{-6-(-4)}$$Simplify the fraction into simplest terms by evaluating the numerator and denominator separately.

$$m=\frac{-5+4}{-6+4}$$Of course, subtracting a negative is the same as adding a positive.

$$m=\frac{-1}{-2}$$The negatives in the numerator and denominator cancel each other out.

$$m=\frac{1}{2}$$

Great! We know the slope! Now, the only variable to figure out next is b, the y-intercept. We can do this by plugging in points of points on the line in the equation.

$$y=\frac{1}{2}x+b$$

In other words, to solve for b, you must plug in a point we know is one the line (either (-6,-5) or (-4,-4)) for x and y. I'll choose (-4,-4):

$$y=\frac{1}{2}x+b$$Plug in the coordinate (-4,-4) in its appropriate spots and then solve for b.

$$-4=\frac{1}{2}*-4+b$$Now, solve for b.

$$-4=-2+b$$Add 2 on both sides.

$$-2=b$$

Now that we have solved for both m and b, the equation that passes through the points (-6,-5) and (-4,-4) is $$y=\frac{1}{2}x-2$$.

Do you need your answer in point-slope form? No problem! Remember the point-slope form

$$y-y_1=m(x-x_1)$$

Of course, m is the slope again. We have already calculated that. Let's substitute that in.

$$y-y_1=\frac{1}{2}(x-x_1)$$

$$y_1\hspace{1mm}\text{and}\hspace{1mm}x_1$$ represent a point on the line. You can either substitute the first or the second set of coordinates. It doesn't matter. However, in the end, your answer should be one of these:

$$y+5=\frac{1}{2}(x+6)$$

$$y+4=\frac{1}{2}(x+4)$$

TheXSquaredFactor Jul 28, 2017