Linear Equations

Linear equation problems will be the problems that deal with some form of \(y=mx+b\).

Example:

A linear function passes through the points (2, 7) and (4, 8). What is the x-intercept of this function?

A) 2

B) 6

C) -6

D) -12

Strategy with coordinates:

  1. Find two coordinate points that are on the line.
    • If the answer choices are all the potential equations, use the points to test them out and find the right one.
  2. Use the slope equation to solve for the slope.
  3. Plug in a point on the line and the slope to \(y=mx+b\).

Strategy with constants:

  1. Get all linear equations into \(y=mx+b\).
  2. Set requested values equal. If the question asks for the slope, and you have a slope value with a constant and one without a constant, set them equal and solve. 

10 Essential Linear Equation Facts

Here are the ten facts about linear equations that you MUST know for the SAT:

  1. linear equation is an equation in which the highest power is 1 (x^1 or just x).
  2. Slope-intercept form is \(y=mx+b\) where is the slope of the line (rise over run), and b is the y-intercept (where the line crosses the y-axis).
  3. In a “real-world” context, the y-intercept represents the starting point, and the slope represents the rate of change.
  4. The slope is constant.
  5. \(\frac{y_1-y_2}{x_1-x_{2_{}}}=m\), where \((x_1,\;y_1)\) and \((x_2,\;y_2)\) are the coordinates of points on the line.
  6. A line with a positive slope runs up and to the right, and a line with a negative slope runs down and to the right.
    • positive slope:
    • negative slope:
  7. The slope of a horizontal line is 0. The slope of a vertical line is undefined.
  8. Parallel lines have the same slope.
  9. Perpendicular lines have opposite reciprocal slopes (like 4, and \(\frac{-1}4\)).
  10. When reflecting a line over the x– or y-axis, simply reverse the sign of the slope (like 4, and -4).

Now that you’ve got those, let’s work through the steps of a traditional linear equation problem:

\(x\)\(y\)
35
1-3

The table above shows two points on line m.  What is the equation of line m?

A) \(\;y=4x-3\)

B) \(\;y=2x+6\)

C) \(y=\frac14x-3\)

D) \(\;y=4x-7\)

First, find two coordinate points.  In this case they would be (3, 5) and (1, -3) and use your coordinate points to find the slope of the line

$$\frac{5-(-3)}{3-1}=\frac82=4$$

Now our equation looks like \(y=4x+b\).  Next, pick any point on the line and plug it in to your unfinished equation to solve for b:

$$5=4\left(3\right)+b\;\;\;\;\;\;b=-7$$

Therefore, our finished equation would be \(y=4x-7\), or answer D).