Linear Equations

Unit Review Sheet

These facts and definitions should be mastered throughout this unit. This page can be used for periodic review and study as you are finishing the unit and in the future.

Facts and Definitions

Lesson 1: Linear Equations With One Variable

To solve an equation:
- First, move the constants and variables to opposite sides.
- Then, isolate the variable.
If the variable is multiplied by a fraction, multiply by the reciprocal to cancel it out. For example, if you have $\frac{2}{5} x = 8$ , multiply both sides by $\frac{5}{2}$ to solve for $x$ .

Lesson 2: Multi-Step Equations

The distributive property allows you to multiply a number across parentheses: a(b+c)=ab+ac.
Follow this process to solve equations:
1. Distribute through parentheses
2. Move the variables and constants to opposite sides
3. Isolate the variable

Lesson 3: How Many Solutions?

The coefficient is the number that multiplies a variable. (e.g., 3 is the coefficient of $3 x$ )
An equation has one solution if simplifying it results in x = a, where a is a single number (e.g., $x = 5$ ).
An equation has infinite solutions if it is true no matter what number is used.
You know an equation has infinite solutions if both sides simplify to the same expression. (e.g., $3 x + 2 = 3 x + 2$ ).
An equation has no solution when no number will make the equation true.
You know an equation has no solution if the simplified version is a false statement. (e.g., $4 x + 5 = 4 x + 8$ or $5 = 8$ ).

Lesson 4: Multi-Step Word Problems

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Lesson 5: Intersection and Graphing

Every equation in the form $y = m x + b$ can be graphed as a straight line.
A system of equations is a set of equations that share the same variables.
The point of intersection (if there is one) is the solution to the system.
A system with one solution means the lines meet (or intersect) at exactly one point.
If two lines have the same slope (parallel lines) but different y-intercepts, there is no solution because they never intersect.

For example, the following system of equations has no solution:
$y = 2 x + 3$
$y = 2 x + 4$
If two equations simplify to the same equation, there are infinite solutions.

For example, the following system has infinite solutions:
$x + 2 y = 4$
$2 x + 4 y = 8$

Lesson 6: Substitution and Elimination

The solution to a system of equations is the point where both equations are true at the same time.
When a system of equations is graphed, any point where the lines intersect is a solution.
To solve a system by graphing, you first rewrite both equations in slope-intercept form, graph the lines, and then find the point where they cross.
To solve a system using substitution, you solve one of the equations for one variable and then substitute that expression into the other equation.
To solve a system using elimination, add or subtract the equations to eliminate one variable.
Subtract the equations if the variable terms have the same coefficient.
Add the equations if the variable terms are opposites.
Elimination with multiples is a strategy when you multiply one (or both) equations first so the coefficients become opposites.

Lesson 7: The Point of It All

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Lesson 8: Linear Algebra In the Wild

Distance = Rate × Time is the key formula for motion problems.
A break-even point is the point where two options cost or take the same amount.
Total Cost = Fixed Fee + (Rate × Quantity)

Lesson 9: Unit 7 Test

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Final Project: Getting Ready for College

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