Writing Linear Equations
Given Slope and a Point

When you are given a real world problem that must be solved, you could be given numerous aspects of the equation. If you are given slope and the y-intercept, then you have it made.

You have all the information you need, and you can create your graph or write an equation in slope intercept form very easily.

However, most times it's not that easy and we are forced to really understand the problem and decipher what we are given. It could be slope and the y-intercept, but it could also be slope and one point or it could be just two points.

If you are given slope and a point, then it becomes a little trickier to write an equation. Although you have the slope, you need the y-intercept.

You have enough information to find the y-intercept, but it requires a few more steps. Let's look at an example.

Example 1: Writing Equations Given Slope and a Point

Write the equation of a line, in slope intercept form, that passes through the point (6, -3) with a slope of -2.

Let's first see what information is given to us in the problem. We'll record this information in the chart below to keep it organized.

Notice that in the chart, the 2 grey sections (slope and y-intercept) are the two numbers that we need in order to write our equation.

We know the slope and a point (x,y). We can use this information to solve for b. Then we can write our equation.

First we will substitute the information that we know (that is in our chart) into the equation, y = mx + b.

Then you will solve for b.

Now we know the slope (m) is -2 because that was given to us.  We also now know the y-intercept (b), which is 9 because we just solved for b.

We can now write our equation!

m = -2

b = 9

y = mx + b

y = -2x + 9

That was a little tougher only because we needed to add the extra step of finding the y-intercept. Since you are so awesome at solving equations, I'm sure this wasn't too painful.

Now let's look at a real world applications of this skill. Here you will have to read the problem and figure out the slope and the point that is given. The slope is going to be your "rate" and the point will be two numbers that are related in some way.

Example 2: Real World Problems Given Slope and a Point

While on vacation in Washington DC, the cab ride from Dulles International Airport to the hotel was 15 miles. The total cost of the trip was $25.50 (excluding tip). The cabbie charges $1.50 per mile for the entire trip. Write a linear equation that can be used to determine the cost of a cab ride to anywhere around Washington DC.

Let's think about what we know in this problem.

• We know that the cabbie charges $1.50 per mile.  I know that this is a rate and therefore, is also the slope. (m = 1.50)
• We also know that 15 miles in the cab costs $25.50. These two numbers are related. This relation means that we know the x and y coordinates of an ordered pair. (15, 25.50)
  

Let's put this information into our chart.

Now, let's substitute these values into our equation, y = mx + b and solve for b.

Now we know the slope (m) is 1.5 because that was given to us.  We also now know the y-intercept (b), which is 3 because we just solved for b.

We can now write our equation!

m = 1.5

b = 3

y = mx + b

y = 1.5x + 3

Ok, so if you are given slope and a point, then you need to substitute for m (slope), x, and y and then solve for b! Once you have m (slope) and b (y-intercept), you can write an equation in slope intercept form.

Now you are ready to solve real world problems given two points. It's not the hard - I promise.

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