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SAT Systems of Equations Word Problems

A large share of SAT system questions arrive as a short story about tickets, snacks, or work hours. Two unknown quantities, two facts about them. The math after setup is routine, so almost all the difficulty lives in translating the sentences into equations and in reading what the final line actually asks for.

The rule

Write one equation counting the items and one equation totaling their value, and label what each letter means with its unit before solving.

Nearly every SAT setup follows the same shape. One sentence counts things and one sentence totals something those things carry, usually money or weight or time. If a store sells 40 items in total made of small boxes and large boxes, that is your count equation: s plus l equals 40. If small boxes cost 3 dollars and large boxes cost 7 dollars and the total was 200 dollars, that is your value equation: 3s plus 7l equals 200.

Define your variables in writing before you do anything else, including the unit. Writing s equals number of small boxes takes two seconds and prevents the most common setup error, which is letting a letter mean the money from small boxes in one equation and the count of them in the other. Once the letters are pinned down, each sentence usually converts directly into one equation.

Solve the count equation for one variable and substitute, since it almost always has coefficients of 1. From s plus l equals 40 you get s equals 40 minus l. Substituting into 3s plus 7l equals 200 gives 120 minus 3l plus 7l equals 200, so 4l equals 80 and l equals 20. Then s equals 20 as well. The arithmetic is short because you picked the easy equation to isolate.

Now read the last line of the question again. It might want the number of large boxes, or the number of small boxes, or the difference between them, or the amount of money the small boxes brought in. These are four different numbers built from the same solution, and the wrong ones are always sitting in the answer choices. Circle the requested quantity before you solve so you do not stop one step early.

Watch for constraints that are not equations at all. Phrases like at least or no more than turn the problem into a system of inequalities rather than equations, and the answer becomes a region or a maximum count instead of a single point. Also keep an eye on units: if one rate is given per hour and a total is given in minutes, convert before you write the second equation.

The trap

Solving the system correctly and then bubbling the variable you solved for when the question asked for a combination such as the sum or the difference.

Try it

Try one
Real question
10203040506020406080Miles drivenTotal charge (dollars)Company ACompany B(40, 60)

The graph shows the total charge, in dollars, for renting a van for m miles from two companies. What is the total charge, in dollars, at the mileage where the two companies charge the same amount?

A jar holds nickels and dimes in the ratio 3 to 4. The coins are worth $5.50 in all. How many dimes are in the jar?

  • A) 10
  • B) 30
  • C) 40 ✓
  • D) 70

Why: Write the counts as 3k nickels and 4k dimes. Their value is 0.05(3k) + 0.10(4k) = 0.55k = 5.50, so k = 10 and there are 40 dimes. Answering 70 gives the total number of coins. Answering 10 means you reported the ratio multiplier rather than the number of dimes. Answering 30 means you reported the number of nickels instead of dimes. Answering 70 means you reported the total number of coins.

Practice more of these free

Questions students ask

How do I decide which two equations to write?

Look for two independent facts about the same pair of unknowns. In most SAT problems one fact is a count or total number of items and the other is a total value in dollars, pounds, or hours. If both of your equations say the same thing in different words, you have not captured two separate constraints.

Is substitution or elimination faster on word problems?

Substitution is usually faster because the count equation has coefficients of 1, so isolating costs nothing. Elimination wins when the question asks for a combination such as x plus y, since adding the equations can produce that combination directly without finding either variable.

Guides that cover this

More on systems of linear equations