SAT Algebra: method and traps

Algebra is the largest single slice of the SAT Math section, and it is the one where speed compounds. Almost every question reduces to the same move made in a different costume: get the unknown alone, or make two expressions equal and see what follows.

Practise the 112 questions Play a timed round

This area at a glance

6 worked examples · 6 traps

112

questions

  • Foundation 27
  • Standard 61
  • Challenge 24

What this area asks for

  • Linear equations in one variable
  • Linear functions and their graphs
  • Systems of two equations
  • Linear inequalities
  • Proportional and direct-variation relationships
  • Function notation

The method

One routine that applies across the whole area, rather than a rule for each question type.

  1. 1

    Name what the question wants before you solve

    A large share of lost marks in this area are not arithmetic errors. They are correct work that answered a different question: solving for x when the question asked for 2x, or for the total when it asked for the difference. Underline the requested quantity first, then solve.

  2. 2

    Clear the clutter, then isolate

    Distribute across parentheses, multiply through by a common denominator to remove fractions, and collect like terms on one side. Only then divide. Doing these in a fixed order means you never have to decide what to do next, which is where time goes.

  3. 3

    For a system, pick the cheaper of substitution and elimination

    If one equation already gives a variable alone, substitute. If the coefficients of one variable match or are easy to match, eliminate. Choosing badly still gets the answer; it just costs a minute you will want later.

  4. 4

    Treat an inequality as an equation with one extra rule

    Every step is the same as for an equation, except that multiplying or dividing by a negative reverses the sign. Circle the negative before you divide, so the flip is a step rather than something you might remember.

Worked examples

Each one is worked to the point where the next move is obvious, not to the answer alone.

1

If 4(x - 3) = 2x + 6, what is the value of x?

  1. 1.Distribute the 4 on the left: 4x - 12 = 2x + 6.
  2. 2.Subtract 2x from both sides to gather the variable: 2x - 12 = 6.
  3. 3.Add 12 to both sides: 2x = 18.
  4. 4.Divide by 2.

x = 9

2

A line passes through (2, 5) and (6, 17). What is its equation in slope-intercept form?

  1. 1.Slope is the change in y over the change in x: (17 - 5) / (6 - 2) = 12 / 4 = 3.
  2. 2.Substitute one point into y = 3x + b. Using (2, 5): 5 = 6 + b.
  3. 3.Solve for the intercept: b = -1.

y = 3x - 1

3

If -2x + 9 > 1, which values of x satisfy the inequality?

  1. 1.Subtract 9 from both sides: -2x > -8.
  2. 2.Divide by -2. Because the divisor is negative, the inequality reverses.
  3. 3.The result is x < 4, not x > 4.

x < 4

4

If 5x + 2y = 24 and 3x - 2y = 8, what is the value of x?

  1. 1.The y terms are opposites, so adding the equations eliminates them: 8x = 32.
  2. 2.Divide by 8 to get x = 4.
  3. 3.Substitute back to check: 5(4) + 2y = 24 gives y = 2, and 3(4) - 2(2) = 8.

x = 4

5

What are the solutions to |2x - 7| = 3?

  1. 1.An absolute value equals 3 when the expression inside is 3 or -3, so solve both.
  2. 2.From 2x - 7 = 3: 2x = 10, so x = 5.
  3. 3.From 2x - 7 = -3: 2x = 4, so x = 2.
  4. 4.Both satisfy the original equation, so both count.

x = 5 and x = 2

6

A taxi charges a fixed 3.50 pounds plus 1.20 pounds a mile. A fare came to 14.30 pounds. How many miles was the journey?

  1. 1.Take off the part that does not depend on distance: 14.30 - 3.50 = 10.80.
  2. 2.What is left was charged at 1.20 pounds a mile, so divide: 10.80 / 1.20.
  3. 3.Check by rebuilding the fare: 3.50 + 9(1.20) = 14.30.

9 miles

Where marks are lost

Each of these has a matching wrong answer waiting among the four choices.

Solving for the wrong thing

When a question asks for 3x rather than x, the value of x will still appear among the choices. Finishing the algebra is not finishing the question.

Forgetting the sign flip

Dividing an inequality by a negative reverses it. This single rule accounts for a large share of wrong answers in the inequality questions, and the un-flipped version is always one of the four choices.

Distributing a subtraction to only the first term

In 5 - 2(x + 4), the -2 multiplies both x and 4, giving 5 - 2x - 8. Writing 5 - 2x + 8 is the most common slip in the whole area.

Assuming a system always has one solution

Two lines with the same slope and different intercepts never meet, and identical equations meet everywhere. A question asking for the value of a constant is often asking you to recognise one of these.

Reading a rate as a total

In C(m) = 45 + 0.12m, the 0.12 is what each extra minute costs and the 45 is what the hire costs before any minutes. Questions ask for one or the other, and the two are almost never both among the choices by accident.

Losing a solution by dividing

Dividing x^2 = 6x by x gives x = 6 and quietly discards x = 0, because dividing by x assumes x is not zero. Move everything to one side and factor instead; the second solution is usually one of the choices.

If you are short on time, spend it here rather than anywhere else in Math. Algebra questions are the most numerous, the most mechanical, and the ones where a reliable routine converts directly into marks.

Now work the questions

112 Algebra questions with a worked explanation for each, on the SAT Math page.

Open SAT Algebra practice

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SAT® Algebra: Method and Practice | IQ Test Center