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Mathematics · 03 / 23

Equations: keep the balance level

Middle school · Worked example and 4 exercises with solutions, at your pace.

Need a foundation?Fractions: share the same wholePowers and roots: count the factors
UnderstandSee an exampleGuided exercise3 progressive challenges

The principle, simply

An equation states that two expressions have the same value. An equivalent transformation applies the same operation to both sides. Adding or subtracting the same number preserves equality, as does multiplying or dividing by a nonzero number. Isolate the unknown without losing solutions.

2*x+3=7
THE PICTURE TO REMEMBEREquations: keep the balance level
  1. 01Equality
  2. 02Same operation
  3. 03Isolate x
  4. 04Substitute to check

WORKED EXAMPLE

From principle to reasoning

2x + 3 = 7. Subtract 3 on both sides: 2x = 4. Divide both by 2: x = 2. Check: 2 × 2 + 3 = 7.

Solve a problem in this subject

  1. Define the objects

    State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.

  2. Choose a transformation

    Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.

  3. Show the reasoning

    Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.

  4. Test and conclude

    Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.

YOUR TURN

01 · Try before checking

Solve 3x − 5 = 10.

One hint

First remove −5 from the left side.

Reveal and understand the solution

Add 5 on both sides: 3x = 15. Divide by 3: x = 5. Substitution gives 15 − 5 = 10.

Compare method, conditions and answer; identify the exact part to revisit.

IMPROVE THROUGH PRACTICE

Three additional challenges with solutions

Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.

Reflect after practising
02 · Recognise and understandLink to this exercise

Solve 5x − 7 = 18 and check.

A hint to get started

Add 7 to both sides.

Step-by-step solution
  1. 5x = 25.
  2. Divide by 5: x = 5.
  3. 5 × 5 − 7 = 18 checks the answer.

Which step did you miss? Retry without help, then explain why the method works.

03 · Apply the methodLink to this exercise

Solve 3(x − 2) = 2x + 4.

A hint to get started

Expand before collecting terms.

Step-by-step solution
  1. 3x − 6 = 2x + 4.
  2. Subtract 2x and add 6: x = 10.
  3. Both sides then equal 24.

Which step did you miss? Retry without help, then explain why the method works.

04 · Go furtherLink to this exercise

Plans cost 12 + 2x and 4x euros for x sessions. When are they equal?

A hint to get started

Translate equal cost into an equation.

Step-by-step solution
  1. 12 + 2x = 4x.
  2. 12 = 2x, so x = 6.
  3. At six sessions both cost €24; below six the second is cheaper.

Which step did you miss? Retry without help, then explain why the method works.

The trap to avoid

Moving a term across a sign hides the operation. State what happens to both sides.

Without notes, can you…

  • Explain the principle in your own words?
  • Rebuild the diagram and its connections?
  • Solve the exercise and justify the method?

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