Algebra (Comprehensive Review)

Chapter 1

Algebra

Course: Sat Prep
Subject: Math
Chapter: Algebra

By the end of this lesson, you will be able to:

• Solve one-variable linear equations.
• Solve multi-step equations.
• Solve equations containing fractions and decimals.
• Solve literal equations.
• Model word problems with equations.
• Apply SAT shortcuts to solve efficiently.

What Is a Linear Equation?

A linear equation is an equation in which the highest power of the variable is 1.

General form:

$ax+b=c$

where:

• $a\neq0$
• $x$ is the variable
• $a,b,c$ are constants

The graph of a linear equation is always a straight line.

Key Vocabulary

| Term | Meaning | | Variable | Unknown quantity | | Constant | Fixed number | | Coefficient | Number multiplying a variable | | Equation | Mathematical statement showing equality | | Solution | Value that makes an equation true |

Rules of Equality

To maintain equality:

• Add the same number to both sides.
• Subtract the same number from both sides.
• Multiply both sides by the same non-zero number.
• Divide both sides by the same non-zero number.

Solving One-Step Equations

Example:

$x+8=17$

Subtract 8 from both sides.

$x=9$

Solving Multi-Step Equations

Example:

$5x-7=18$

Add 7.

$5x=25$

Divide by 5.

$x=5$

Variables on Both Sides

Example:

$4x+5=2x+17$

Subtract $2x$.

$2x+5=17$

Subtract 5.

$2x=12$

Divide by 2.

$x=6$

Fractions in Equations

Example:

$\frac{x}{4}+3=10$

Subtract 3.

$\frac{x}{4}=7$

Multiply by 4.

$x=28$

Literal Equations

Example:

$A=lw$

Solve for $l$.

Divide both sides by $w$.

$l=\frac{A}{w}$

Formula Summary

| Formula | Use | | $ax+b=c$ | Standard linear equation | | $y=mx+b$ | Slope-intercept form | | $m=\frac{y_2-y_1}{x_2-x_1}$ | Slope formula |

Graph of a Linear Equation

Students would be shown an interactive graph such as:

• slope
• y-intercept
• increasing vs. decreasing lines
• horizontal and vertical lines

Example 1

Solve:

$7x-9=26$

Solution

Add 9.

$7x=35$

Divide by 7.

$x=5$

Example 2

Three times a number increased by 8 equals twice the number plus 20.

Find the number.

Let the number be $x$.

$3x+8=2x+20$

Subtract $2x$.

$x+8=20$

Subtract 8.

$x=12$

SAT Strategy

Instead of fully solving, sometimes substitute the answer choices back into the equation to save time, especially on multiple-choice questions.

Common Mistakes

• Forgetting to distribute negative signs.
• Performing different operations on each side of the equation.
• Incorrectly combining unlike terms.
• Arithmetic errors with fractions.

Desmos Calculator Tip

Use the built-in Desmos graphing calculator to graph both sides of an equation (e.g., $y=3x+8$ and $y=2x+20$). The x-coordinate of the intersection gives the solution.

Lesson Summary

• Linear equations have variables with an exponent of 1.
• Solve by performing the same operation on both sides.
• Check solutions by substitution.
• Graphs of linear equations are straight lines.