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:**

**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.**

**Use the slope equation to solve for the slope.****Plug in a point on the line and the slope to**\(y=mx+b\).

**Strategy with constants:**

**Get all linear equations into**\(y=mx+b\)**.****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:

- A
*linear equation*is an equation in which the highest power is 1 (*x*^1 or just*x*). **Slope-intercept form**is \(y=mx+b\) where*m*is the slope of the line (rise over run), and*b*is the*y*-intercept (where the line crosses the*y*-axis).- In a “real-world” context, the
*y*-intercept represents the starting point, and the slope represents the rate of change. - The slope is constant.
- \(\frac{y_1-y_2}{x_1-x_{2_{}}}=m\), where \((x_1,\;y_1)\) and \((x_2,\;y_2)\) are the coordinates of points on the line.
- 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:

- positive slope:
- The slope of a horizontal line is 0. The slope of a vertical line is undefined.
- Parallel lines have the same slope.
- Perpendicular lines have opposite reciprocal slopes (like 4, and \(\frac{-1}4\)).
- 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:

\(x\) | \(y\) |
---|---|

3 | 5 |

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).

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