Linear Equations from Word Problems

To excel at translating word problems, you must recognize that every linear scenario is built from a few fundamental components. On the SAT, these usually take the form of the slope-intercept structure, even if the question only involves one variable.

The Anatomy of a Linear Equation

Most linear word problems can be modeled by the equation: $Total = (Rate \times Variable) + Constant$ In algebraic terms, this is our familiar: $y = mx + b$

• The Constant ($b$): This is the "starting" or "initial" value. In a word problem, look for words like flat fee, upfront cost, initial deposit, base price, or at the beginning. This value does not change regardless of the variable.
• The Rate of Change ($m$): This is the "slope." Look for words like per, each, every, hourly, daily, or for every. This value is always attached to the variable.
• The Variable ($x$): This represents the quantity that changes, such as number of hours, miles driven, or items purchased.

The Translation Table

Success on the SAT requires an instant "dictionary" for translating English to Math:

• "Is", "Was", "Will be", "Results in": Equals ($=$)
• "Of", "Times", "Product": Multiplication ($\times$)
• "Per", "Each", "Every": Multiplication (usually indicating the slope $m$)
• "More than", "Increased by", "Sum": Addition ($+$)
• "Less than", "Decreased by", "Difference": Subtraction ($-$)
• "Ratio", "Quotient", "Out of": Division ($\div$)

Equations with Variables on Both Sides

Harder SAT questions often involve comparing two different scenarios to find when they are equal. For example, comparing two different gym memberships to find after how many months the total cost is the same. The structure for these is: $m_1x + b_1 = m_2x + b_2$ Where $m_1$ and $b_1$ are the rate and constant for the first scenario, and $m_2$ and $b_2$ are for the second.

Unit Consistency

One of the most common "Hard" level traps is a mismatch in units. If a rate is given in dollars per hour but the time is given in minutes, you must convert the units before setting up your equation. $Rate \times \left(\frac{Minutes}{60}\right) = Total$ Always check the units of your variable against the units of your rate.

Reference Sheet vs. Memory

Note: The SAT Reference Sheet does not contain the formula for linear equations or any translation tips. You must memorize the $y = mx + b$ structure and the translation keywords listed above. The reference sheet is primarily for geometry and trigonometry, making your mastery of these algebraic concepts even more vital.

1. Read the Last Sentence First

SAT word problems are often long. Before reading the story, look at the very last sentence to identify exactly what the question is asking for. Are you solving for $x$? Are you solving for the total cost $y$? Or are you solving for a specific expression like $2x - 10$? Knowing the goal prevents you from doing unnecessary work.

2. The "Labeling" Method

As you read, physically label the numbers in the text. Write a small "$m$" over rates and a small "$b$" over initial values. This turns a paragraph of text into a list of components, making the equation setup nearly automatic.

3. Use Desmos Strategically

On the Digital SAT, the Desmos calculator is your best friend. For "Hard" linear equations, you can often type the equation exactly as you wrote it into the calculator.

• If you have an equation like $1.50x + 20 = 2.25x + 5$, type it into line 1. Desmos will show a vertical line at the solution for $x$.
• If you are looking for an intersection, type both sides as separate equations ($y = 1.50x + 20$ and $y = 2.25x + 5$) and click the point where they cross.

4. Backsolving (Plugging in Answers)

If the question asks for a single value and you are struggling to set up the equation, use the answer choices. Start with choice (B) or (C). Plug the value into the scenario described. If the result is too small, try a larger number; if it's too big, try a smaller one.