SAT Math is heavy on linear equations. Knowing this well gives you the basis for the proper set up and solution of the numerous word problems on SAT Math. So let’s go…
What is a linear equation?
A linear equation is an equation which when plotted makes a line. Below are some of the examples of plotted linear equations.
Slope y-intersept form
It’s easier to plot an equation of a line on the x-y plane, if it’s in a slope y-intercept form.
y = mx + b
Where m is the slope and b is the point at which the line intercepts y-axis.
The slope is defined as the change in y coordinate divided by the change in the x coordinate, or simply put rise over run.
Let’s do some examples.
Let’s look at a line y = 3x+1 on the graph:
Based on the equation, its slope is 3. Let’s see if we can see it on the graph.
Let’s consider 2 points on the line, (-1,-2) and (o,1).
When we count the number of units directly up from the bottom point to the next point, along the y-axis, it’s 3.
The difference between the two points mathematically confirms the same: 1-(-2)=3.
So our rise is 3 points up, and up is positive direction.
When we count the number of units over to the side from the point (-1,-2) to the point (0,1), it’s 1 unit to the right, which is also in the positive direction.
When we do the difference between point 0 and -1 mathematically, we get 1 as well, 0-(-1)=1. So our run is 1 unit to the right.
So when we calculate the rise over the run, or the difference in y divided by the difference in x, we get 3 (3/1=3). So our slope confirms to be 3.
Let’s now look at the y-intercept. Based on the equation y=3x+1, the y-intercept is 1.
Does the graph confirm that? Yes, it intercepts y at a point (0,1). We can also prove it algebraically.
What is point x when the line intercepts y-axis?
The point x is always 0 when the line intercepts y-axis.
So plugging 0 in for x in the equation we get
y = 3*0+1 or
y=1
So when x=0, a line intercepts y-axis. And the intercept can be found directly from the equation of the line in the slope y-intercept form.
Lets do some more examples
Can you figure out the equation of a line shown on the graph ? 
We need to produce an equation of the form y= mx+b
First lets find the slope m. We locate the two points (-1,-1) and (0,4).
We can see that to get from (-1,-1) to (0,4) vertically, it takes 5 units up, so our rise is 5.
To get from (-1,-1) and (0,4) horizontally, it takes 1 unit to the right, so our run is 1.
If the slope is rise over run, then it’s 5/1, which is 5.
Now let’s find b, which is the y-intercept. Looks like the line crosses y-axis at the point (0,4), so the y-intercept is 4.
Now we can write the equation of this line:
y = 5x+4.
Now try graphing this one on your own, and if you want to check your answer, watch the video below! If you need more help in making or graphing linear equations or to prepare for SAT Math test, book a one-on-one session with me or use the contact form above to contact me.
