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Algebra / LESSON 01

Linear equations · SAT Math

10 min + practice · Original foundation lesson

An equation is a balance. Applying the same reversible operation to both sides keeps its solutions unchanged.

A reliable method

Distribute, combine like terms, then collect variable terms on one side. Substitute your answer into the original equation.

The common trap

Dividing only one term on a side changes the equation. Divide the entire side, and distribute a negative sign to every term.

Make the connection

An expression such as 3x + 2 may be the requested answer. Read the last sentence before solving for x.

Compare two valid routes

For 3(x+4)=27, route one divides the whole equation by 3, giving x+4=9 and x=5. Route two distributes first: 3x+12=27, then 3x=15 and x=5. Both routes preserve equality because every term on the affected side receives the operation. Dividing first is shorter here; distributing first helps when there are variable terms outside the bracket. Check 3(5+4)=27 before moving on.

Diagnose a tempting error

A learner writes 3x+4=27 from 3(x+4)=27. The constant 4 also lies inside the brackets, so distribution must produce 12. Their answer 23/3 fails substitution into the original equation. If the final question asks for 2x+1, the correct x=5 is only an intermediate result: the requested answer is 11.

Try, then check

Solve 4(x−2)=3x+7, then find x−5. First try without expanding both sides at once. Check: 4x−8=3x+7 gives x=15, and x−5=10. Substitution gives 52 on each side. Explain why x=15 is not the final requested answer.

WORKED EXAMPLE

If 3x + 4 = 19, what is the value of x?

  1. Subtract 4 from both sides: 3x = 15.
  2. Divide both sides by 3: x = 5.
  3. Check: 3(5) + 4 = 19.
Answer: 5
Compare the other answer choices
5/3

Divided by the coefficient twice.

19/3

Divided the total by the coefficient before removing the constant.

23/3

Added the constant instead of subtracting it.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

Official mathematics skillsSiyavula Grade 10 mathematics