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CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

A linear equation is an algebraic equation that, when graphed, creates a straight line. The point-slope form is one way to write a linear equation. This lesson will teach you how to write a linear equation in point-slope form and give you some examples.

A **linear equation** is made up of one or more terms that are either constants or the product of a constant and a single variable (such as 2*x*). The terms of the variable must be to the single power and not squared, cubed, or more, but the equation can have more than one variable.

Linear equations have many practical uses; they're used extensively in banking and finance and can be used to help with personal finances. They also have scientific and engineering applications.

Here are some **examples of linear equations**:

*y* = 2*x* + 5

3*m* - 2*n* = 6

*a*/2 = *b* + 1

Here are some **examples of non-linear equations**:

*y*^2 = *x* + 2

âˆš5*x* - 2*y* = 7

4/*x*^2 + 2*y* + 3 = 0

There are many different ways to write a linear equation, including:

**Slope-intercept form**:*y*=*mx*+*b*. This is the most common form of a linear equation, where*m*is the slope and*b*is the*y*-intercept.**Point-slope form**: (*y*-*y*1) =*m*(*x*-*x*1)**General form**:*Ax*+*By*+*C*= 0 (*A*and*B*cannot both be zero)

In point-slope form (which is written like this: (*y* - *y*1) = *m*(*x* - *x*1)), *y*1 is the *y* value of the known point on the line, *m* is the slope, and *x*1 is the *x* value of the known point.

This form of a linear equation is derived from the equation for finding the slope of a line. The **slope of a line** is the ratio of the line's elevation to its horizontal movement, or as it's more commonly known, rise over run.

The **equation for finding the slope of a line** is:

*m* = (*y*2 - *y*1)/(*x*2 - *x*1)

You can rearrange this formula by multiplying both sides of the equations by (*x*2 - *x*1) to get:

(*y*2 - *y*1) = *m*(*x*2 - *x*1)

The point-slope form of a linear equation is most useful for finding a point on a line when you know the slope and one other point on the line. It can also be used to find a point on the line when you know two other points.

Let's now go over some example problems to help cement everything we've covered.

A line goes through point (1, 3) with a slope of 2. What is the equation of the line? Start by using the point-slope formula to find the equation.

(*y* - *y*1) = *m*(*x* - *x*1)

*y* - 3 = 2(*x* - 1)

Then simplify.

*y* - 3 = 2*x* - 2

*y* = 2*x* + 1

What is the equation of a line that goes through point (-2, 4) and the origin? The first step is to find the slope of the line.

*m* = (*y*2 - *y*1)/(*x*2 - *x*1)

It doesn't matter which point is 1 or 2; just remember to keep it consistent throughout the problem.

*m* = (4 - 0)/(-2 - 0)

*m* = 4/-2 = -2

So, the slope of the line that runs through the origin and point (-2, 4) on the coordinate plane is -2.

The next step is to find the equation for that line. To do this, plug the numbers you know into the point-slope formula. Again, it does not matter which point you use, just as long as you are consistent. Whenever possible, use the origin because the zeros make it easier to simplify the equation.

(*y* - *y *1) = *m* (*x* - *x*1)

*y* - 0 = -2(*x* - 0)

*y* = -2*x*

Just to show you that either point will work, let's plug in the point (-2, 4) and see what happens.

*y* - (4) = -2(*x* - (-2))

*y* - 4 = -2*x* - 4

*y* = -2*x*

Say you are planning a fundraiser. The data for past fundraisers reveals that when 700 people attended the fundraiser, the group made $1000, and when 1000 people attended the fundraiser, the group made $1500. Your goal is to make $2500. How many people will need to attend your fundraiser to reach your goal? (Assume a linear relationship.)

This problem adds one more step to the end of the problem, but the first thing we need to do is find the equation for the linear relationship. Since we know two points (attendees and money made), we can use those points to find the equation.

First, determine the slope of the line using the points (700, $1000) and (1000, $1500).

*m* = (1500 - 1000)/(1000 - 700)

*m* = 500/300 = 1.6

Now, use the slope to write an equation:

(*y* - *y*1) = *m* (*x* - *x*1)

*y* - 1000 = 1.6(*x* - 700)

*y* - 1000 = 1.6*x* - 1120

*y* = 1.6*x* - 120

Now, for the final step: Since we know that the fundraising goal is $2500, you can plug that in for *y* in the equation. It replaces *y*, because that is where the money was in all the other points used so far in this example.

2500 = 1.6*x* - 120

1.6*x* = 2620

*x* = 1637.5

The **point slope form of a linear equation** is (*y* - *y*1) = *m*(*x* - *x*1) and is useful for finding the equation for a line when you know one point along the line and the slope of the line. It's derived from the equation for finding the slope of a line and has practical uses in many areas of mathematics and the real world.

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CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

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