OMPT drill — Algebra

OMPT linear equations practice: lesson and 10 problems

Isolate the unknown, keep both sides honest. Sounds trivial until a fraction or a bracketed negative shows up; that is where I watch students drop easy points. Work through all ten below before moving on; algebra slips compound into every other strand.

Tested inOMPT-A OMPT-B OMPT-C OMPT-D OMPT-E OMPT-F OMPT-G

OMPT linear equations: the lesson

What the OMPT tests in linear equations

A linear equation is a balance with one unknown in it. Whatever sits on the left equals whatever sits on the right, and every legal move keeps that true: add the same thing to both sides, subtract it, multiply or divide both sides by the same non-zero number. That is the entire theory. Solving $3 x - 5 = 7$ means undoing what was done to $x$ , in reverse order: the 5 was subtracted last, so add it back first ( $3 x = 12$ ), then divide by 3 ( $x = 4$ ).

Because the theory is so small, the OMPT does not test whether you know it. It tests whether you can run it without slipping when the equation is dressed up: brackets to expand, fractions to clear, the unknown on both sides at once, a parameter standing in for a number. The chapter "Linear formulas and equations" sits on every variant from OMPT-A to OMPT-G and feeds the word problems and the systems chapter too. Questions are phrased plainly: "Solve for $x$ ", "For which value of $p$ has the equation no solution", or a two-sentence story that hides one equation. Nearly every wrong answer I mark here is a sign error or a step done to one side only. The skill you are training is bookkeeping.

Notation and answer format the OMPT expects

You type the answer into a box, and the platform wants the exact value. Keep these habits from the first drill onward.

Fractions stay fractions: type $19/5$ , not $3.8$ , unless the question says "round to". Simplify them.
A negative answer needs its sign in the box: $- 11$ . Half of the marks lost on this topic are a dropped minus.
If the unknowns cancel, the answer is a phrase, not a number: "no solution" when a false statement remains, "all real numbers" (every $x$ ) when a true one does.
With a parameter, give $x$ as an expression in that parameter, for example $x = \frac{a + 3}{a - 2}$ , and state the value the parameter may not take.
Word problems want the quantity asked for, in the unit asked for: minutes if it says minutes.

Worked example 1: brackets and the unknown on both sides

Solve $3 (x - 4) = 7 - 2 x$ .
Expand the left side: $3 x - 12 = 7 - 2 x$ .
Add $2 x$ to both sides to gather the unknowns: $5 x - 12 = 7$ .
Add 12 to both sides: $5 x = 19$ , so $x = \frac{19}{5}$ .
Check it, because the check is cheap: left side $3 (\frac{19}{5} - 4) = 3 \cdot (- \frac{1}{5}) = - \frac{3}{5}$ ; right side $7 - \frac{38}{5} = - \frac{3}{5}$ . They match. Type $19/5$ , not 3.8.

Worked example 2: clearing fractions

Solve $\frac{x}{3} + \frac{x - 1}{4} = 2$ .
The denominators are 3 and 4, so multiply every term by 12. The 2 on the right gets multiplied too; forgetting it is the classic slip here.
That gives $4 x + 3 (x - 1) = 24$ .
Expand and collect: $4 x + 3 x - 3 = 24$ , so $7 x = 27$ and $x = \frac{27}{7}$ .
Quick check: $\frac{27}{21} + \frac{20/7}{4} = \frac{9}{7} + \frac{5}{7} = \frac{14}{7} = 2$ . Correct. The trap was the lonely constant: multiply it by 12 like everything else.

The full lesson works through the three-move routine, equations with letters as coefficients, the two special endings (no solution and infinitely many), fractions with the unknown in the denominator, and two word problems of the kind the OMPT-A and OMPT-E use. It closes with the slips I see most often and how to catch each one in ten seconds.

How each OMPT variant tests linear equations

The chapter weights are the official ones from the omptest.org learning-assessment pages. The question counts are my estimates of how much of that chapter this one topic takes up. The 21-question papers (OMPT-D and OMPT-F) ask long multi-part questions, so a topic there usually shows up as a step inside a bigger problem.

OMPT-A
52 questions
Testedroughly 5 to 7 of 52 questions
Official chapter: Linear formulas, equations and systems of linear equations, 18% of the paper.
Short numeric answers. Brackets, fractions, the unknown on both sides, one equation with a parameter, and a two-sentence word problem.
OMPT-B
64 questions
Testedroughly 5 to 8 of 64 questions
Official chapter: Linear formulas, equations and systems of linear equations, 8% of the paper.
Same shapes as OMPT-A plus a "how many solutions" question with a parameter, and linear steps hidden inside later calculus and trig items.
OMPT-C
30 questions
Testedroughly 4 to 6 of 30 questions
Official chapter: Linear formulas and equations, 25% of the paper.
With only four chapters on the paper, linear equations carry more weight: expect fractions to clear, a formula to rearrange, and a word problem.
OMPT-D
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, equations and inequalities, 30% of the paper.
Rarely a standalone item. A linear equation appears as the last step of a derivative, tangent or geometry question, often with a parameter.
OMPT-E
40 questions
Testedroughly 3 to 5 of 40 questions
Official chapter: Linear formulas, equations and systems of linear equations, 10% of the paper.
Numeric answers and story problems (rates, prices, mixtures) that translate into one linear equation. Systems of two equations also appear.
OMPT-F
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Equations and inequalities, 30% of the paper.
Multi-part questions where a linear equation with a parameter finishes a functions or differentiation problem. Rearranging a formula for one letter is common.
OMPT-G
26 questions
Testedroughly 2 to 4 of 26 questions
Official chapter: Functions, graphs and equations, 45% of the paper.
Linear equations arrive through physics and chemistry: rearrange a formula, solve for a time or a concentration, give the number with its unit.

OMPT linear equations practice problems

10 original problems in the formats the OMPT uses: typed numeric answers, multiple choice, and spot-the-error. The last few are harder than the exam average on purpose. Solutions and the common mistake sit behind each problem.

Problem 1Numeric answer

Solve for $k$ : $3 (2 k - 5) = k + 10$ .

Show the worked solution

Answer: 5

I start by expanding the left side: $6 k - 15 = k + 10$ . Then I move the $k$ -terms to one side, $5 k = 25$ , and divide by 5 to get $k = 5$ . A quick check: $3 (10 - 5) = 15$ and $5 + 10 = 15$ , so it holds.

Problem 2Multiple choice

For which value of $p$ does the equation $p x + 4 = 2 x + 7$ have no solution at all?

A $p = 2$
B $p = - 2$
C $p = \frac{7}{4}$
DNo such value exists

Show the worked solution

Answer: A — $p = 2$

I rewrite the equation as $(p - 2) x = 3$ . If $p \neq = 2$ I can divide and get exactly one solution. But when $p = 2$ the left side collapses to $0 = 3$ , which is false for every $x$ — no solution. So $p = 2$ is the answer.

Problem 3Numeric answer

A phone plan costs a fixed $€ 12$ per month plus $€ 0.08$ per minute of calling. In one month the bill came to $€ 19.20$ . How many minutes were called?

Show the worked solution

Answer: 90

I let $m$ be the minutes and write the bill as $12 + 0.08 m = 19.20$ . Subtracting 12 gives $0.08 m = 7.20$ , and dividing by 0.08 gives $m = 90$ minutes. I always subtract the fixed part first — the per-minute rate only applies to the variable part of the bill.

Problem 4Multiple choice

Solve for $m$ : $\frac{m - 3}{4} = \frac{2 m + 1}{6}$ .

A $m = - 11$
B $m = 11$
C $m = - 7$
D $m = \frac{7}{2}$

Show the worked solution

Answer: A — $m = - 11$

I clear both denominators by multiplying through by 12: $3 (m - 3) = 2 (2 m + 1)$ , so $3 m - 9 = 4 m + 2$ . Collecting terms gives $- 11 = m$ . Checking: $(- 11 - 3) /4 = - 3.5$ and $(- 22 + 1) /6 = - 3.5$ . It matches.

Problem 5Spot the error

A student solves $5 - 2 t = 11$ like this. Step 1: $- 2 t = 11 - 5$ . Step 2: $- 2 t = 6$ . Step 3: $t = 3$ . Which step contains the error?

Show the worked solution

Answer: Step 3

Steps 1 and 2 are fine: subtracting 5 from both sides gives $- 2 t = 6$ . The slip is in Step 3. Dividing 6 by $- 2$ gives $t = - 3$ , not $t = 3$ — the student dropped the minus sign on the coefficient. Substituting $t = - 3$ back in: $5 - 2 (- 3) = 5 + 6 = 11$ . Correct.

Problem 6Numeric answer

Two numbers add up to 26. The larger number is 5 more than twice the smaller. What is the larger number?

Show the worked solution

Answer: 19

I call the smaller number $x$ , so the larger is $2 x + 5$ . Their sum gives $x + 2 x + 5 = 26$ , so $3 x = 21$ and $x = 7$ . The larger number is then $2 \cdot 7 + 5 = 19$ . I check the sum: $7 + 19 = 26$ .

Problem 7Numeric answer

Solve for $y$ : $2 (3 y + 1) - 5 (y - 2) = 4 - 3 (y - 6)$ .

Show the worked solution

Answer: 2.5

Brackets first, all of them. Left side: $6 y + 2 - 5 y + 10 = y + 12$ . Right side: $4 - 3 y + 18 = 22 - 3 y$ . So $y + 12 = 22 - 3 y$ , which collects to $4 y = 10$ and $y = \frac{5}{2} = 2.5$ . Check: left $2 (8.5) - 5 (0.5) = 17 - 2.5 = 14.5$ , right $4 - 3 (- 3.5) = 4 + 10.5 = 14.5$ . Equal.

Problem 8Multiple choice

Solve $\frac{2 x + 1}{3} - \frac{x - 2}{5} = 3$ .

A $x = \frac{34}{7}$
B $x = \frac{46}{7}$
C $x = - \frac{8}{7}$
D $x = 4$

Show the worked solution

Answer: A — $x = \frac{34}{7}$

The common denominator is 15, so I multiply every term by 15, the 3 on the right included: $5 (2 x + 1) - 3 (x - 2) = 45$ . Expand carefully: $10 x + 5 - 3 x + 6 = 45$ , so $7 x = 34$ and $x = \frac{34}{7}$ . Check: $\frac{2 \cdot 34/7 + 1}{3} = \frac{75/7}{3} = \frac{25}{7}$ and $\frac{34/7 - 2}{5} = \frac{20/7}{5} = \frac{4}{7}$ ; the difference is $\frac{21}{7} = 3$ .

Problem 9Numeric answer

The equation $(m^{2} - 1) x = m + 1$ has infinitely many solutions for exactly one value of $m$ . Give that value.

Show the worked solution

Answer: -1

Infinitely many solutions needs the form $0 \cdot x = 0$ , so both the coefficient and the right side must vanish. The coefficient factors: $m^{2} - 1 = (m - 1) (m + 1)$ , zero for $m = 1$ and $m = - 1$ . Test each. With $m = 1$ : $0 \cdot x = 2$ , false for every $x$ , no solution. With $m = - 1$ : $0 \cdot x = 0$ , true for every $x$ , infinitely many. So $m = - 1$ . For any other $m$ there is exactly one solution, $x = \frac{1}{m - 1}$ ; check $m = 3$ : $8 x = 4$ , $x = \frac{1}{2}$ , and $\frac{1}{3 - 1} = \frac{1}{2}$ .

Problem 10Numeric answer

A cyclist leaves town at 9:00 riding at a steady 18 km/h. At 9:40 a car leaves the same town along the same road at 54 km/h. How many minutes after 9:40 does the car catch the cyclist?

Show the worked solution

Answer: 20

Let $t$ be the time in hours after 9:40. By then the cyclist has already ridden 40 minutes, which is $\frac{2}{3}$ of an hour, so its distance is $18 (t + \frac{2}{3}) = 18 t + 12$ . The car has gone $54 t$ . Catching up means equal distances: $54 t = 18 t + 12$ , so $36 t = 12$ and $t = \frac{1}{3}$ hour, which is 20 minutes. Check: in 20 minutes the car covers $54 \cdot \frac{1}{3} = 18$ km, and the cyclist has ridden exactly one hour since 9:00, also 18 km.

OMPT linear equations: questions students ask

Are linear equations on the OMPT-A?

Yes. "Linear formulas and equations" is one of the eight OMPT-A chapters, so roughly 5 to 7 of the 52 questions come from it, and more use it as a step. Expect brackets, fractions, the unknown on both sides, one equation with a parameter, and a short word problem. Systems of two linear equations are a separate chapter on the same paper.

How do I solve a linear equation with fractions for the OMPT?

Multiply every term on both sides by the common denominator, then expand brackets and collect. For $\frac{x}{3} + \frac{x - 1}{4} = 2$ the multiplier is 12, giving $4 x + 3 (x - 1) = 24$ , so $x = \frac{27}{7}$ . The usual slip is forgetting to multiply the constant on the right. Give the exact fraction as your answer, not a decimal.

What does "no solution" mean for a linear equation on the OMPT?

If the unknown cancels from both sides and a false statement remains, such as $3 = 7$ , no value of $x$ works and the answer is "no solution". If a true statement remains, such as $0 = 0$ , every real number is a solution. The OMPT asks this with a parameter: find the value that makes the coefficient of $x$ zero, then check which of the two cases you are in.

Do I need a calculator for linear equations on the OMPT?

No, and none is provided on the maths variants unless a question supplies one. The numbers are chosen so that fractions stay small. If you find yourself dividing 19.20 by 0.08 by hand, multiply both sides by 100 first and work with integers. Decimal answers are only expected when the question says to round.

How many linear equation questions are on the OMPT-C?

The OMPT-C has 30 questions over four chapters (numbers, algebra, linear formulas and equations, geometry), so linear equations are a large slice, roughly 4 to 6 questions plus the ones hidden inside algebra and geometry items. Rearranging formulas, clearing fractions and translating a word problem are the shapes to drill.

What is the difference between a linear equation and a linear inequality on the OMPT?

The moves are the same, with one exception: when you multiply or divide an inequality by a negative number the sign flips. $- 2 x < 6$ becomes $x > - 3$ . An equation ends in a number; an inequality ends in a region, written as $x > - 3$ or as an interval. The OMPT tests both, and the flip is where the marks go missing.

Test hubs that cover linear equations:OMPT-A OMPT-B OMPT-C OMPT-D OMPT-E OMPT-F OMPT-G All 28 topics

OMPT linear equations practice: lesson and 10 problems

OMPT linear equations: the lesson

What the OMPT tests in linear equations

Notation and answer format the OMPT expects

Worked example 1: brackets and the unknown on both sides

Worked example 2: clearing fractions

How each OMPT variant tests linear equations

OMPT linear equations practice problems

OMPT linear equations: questions students ask

Related OMPT topics and test hubs