Mastering Linear Equations in One Variable Questions on the SAT

Algebra problems comprise ~35% of the questions on the SAT math section, and one of the most common types of algebra questions is linear equations in one variable. At their core, these equations are simple enough, structured around the basic principle of finding the value of a single variable, usually denoted as $x$.

However, the simplicity is deceptive. These equations can easily become complex, involving fractions and parentheses and requiring numerous steps to solve. The difficulty escalates when equations are compounded with extraneous terms or require manipulations that aren’t immediately apparent.

Solving single-variable linear equation problems isn't just about finding the right answer; it's about understanding the logical process that leads to that answer. By following our 4-step process below, you will learn to systematically solve these problems every time, no matter how complex they seem.

Step 1: Simplify Each Side of the Equation

The first step is to streamline the equation by reducing complexity through algebraic manipulations, such as combining similar terms and resolving parentheses, which is crucial for making the equation more manageable and clearer to solve.

Combine Like Terms

Add or subtract terms with the same variable simplify the equation.

For example: $5x + 3 - 2x + 4 = 10$ simplifies to $3x + 7 = 10$

Eliminate Parentheses

Use the distributive property to multiply terms outside the parentheses with those inside, simplifying the equation further.

For instance: $2(3x + 4) - 5 = 11$ becomes $6x + 8 - 5 = 11$ and further simplifies to $6x + 3 = 11$

Simplify Complex Fractions

By finding a common denominator and multiplying all terms by it, you can eliminate fractions and make the equation more straightforward.

An example is: $\frac{x}{4} + \frac{3}{6} = \frac{2}{3}$ which simplifies to $3x + 6 = 8$

Rearrange the Equation

Organize the equation for clarity, often by moving all terms with the variable to one side and constants to the other.

For example: $x + 5 = 3x - 7$ rearranges to $-2x = -12$

Deal with Special Cases

Address unique situations like products equalling zero or variables in denominators, which require specific approaches.

For example: $x(x - 5) = 0$ leads to solutions $x = 0$ or $x = 5$. Another example is $\frac{1}{x} + 2 = 5$ which simplifies to $1 + 2x = 5x$

Complete Example

Let's take an equation and apply the simplification steps:

$3(x + 2) - 2x = 5 + x - 1$

First, eliminate the parentheses and combine like terms:

$3x + 6 - 2x = 5 + x - 1$

Then, simplify the equation further by combining all like terms:

$3x - 2x - x = 5 - 1 - 6$

The equation becomes:

$-x = -2$

Step 2: Get the Variable on One Side

This step involves manipulating the equation to isolate the variable on one side, which is key to solving the equation and finding the value of the variable.

Isolate the Variable

Use addition, subtraction, multiplication, or division to get the variable by itself on one side of the equation.

For example: $5x - 3 = 12$ can be simplified by adding 3 to both sides, resulting in $5x = 15$

Undo Addition or Subtraction

If there’s a number added or subtracted from the variable, do the opposite operation on both sides of the equation to isolate the variable.

For example: $x + 4 = 9$ becomes $x = 9 - 4$, which simplifies to $x = 5$

Undo Multiplication or Division

If the variable is multiplied or divided by a number, reverse the operation on both sides of the equation to get the variable alone.

For instance: $4x = 20$ can be simplified by dividing both sides by 4, leading to $x = \frac{20}{4}$, which simplifies to $x = 5$

Complete Example

Let’s apply these steps to another equation:

$3x + 7 = 10$

First, undo the addition by subtracting 7 from both sides:

$3x = 10 - 7$

Then, undo the multiplication by dividing each side by 3:

$x = \frac{3}{3}$

The equation simplifies to:

$x = 1$

Step 3: Simplify the Equation Again

After isolating the variable, this step involves further simplification to ensure the equation is in its simplest form, facilitating an easier and more accurate solution.

Further Simplification

Check if the equation can be simplified further by combining like terms or performing basic arithmetic operations.

For example: $2x + 3x = 10$ can be simplified to $5x = 10$

Reduce Fractions

If your solution is a fraction, reduce it to its simplest form.

For instance: $\frac{8}{4} = x$ simplifies to $2 = x$

Decimal and Percentage Conversion

Convert decimals to fractions or percentages if it makes the equation simpler or more understandable.

Example: $x = 0.75$ can be expressed as $x = \frac{3}{4}$ or $x = 75\%$

Complete Example

Let’s apply these steps to simplify an equation:

$4x + 2x - x = 12$

Combine like terms:

$5x = 12$

If possible, reduce to simpler form:

$x = \frac{12}{5}$

Step 4: Check Your Solution

This crucial step ensures that the solution you've found actually satisfies the original equation, verifying the accuracy of your work.

Substitute the Solution Back Into the Original Equation

Replace the variable with your solution in the original equation to see if the equation holds true.

For example, if the solution is $x = 3$ for the equation $2x + 4 = 10$, substitute ‘3’ for ‘x’ to check: $2(3) + 4 = 10$

Verify the Equality

Perform the necessary arithmetic to confirm that both sides of the equation are equal after the substitution.

Continuing the example: $6 + 4 = 10$ confirms that the solution is correct.

Complete Example

Let's verify the solution for the equation:

$3x + 4 = 13$, where we found $x = 3$ as the solution.

Substitute '3' for 'x' in the original equation:

$3(3) + 4 = 13$

Perform the arithmetic:

$9 + 4 = 13$ shows that the solution is indeed correct.