IB MYP 3 Mathematics 4.2 Equations with Variables on Both Sides and Rational Coefficients Study Notes - New Syllabus

4.2 – Equations with Variables on Both Sides and Rational Coefficients

In the previous section, the variable appeared on only one side of the equation. In this section, the variable may appear on both sides, and the coefficients may be rational numbers, including fractions, integers, and terminating decimals.

The main goal remains the same: isolate the variable on one side while maintaining the balance of the equation.

The textbook emphasizes the same fundamental principle: when an unknown appears more than once, we perform a sequence of operations on both sides until the unknown is isolated. :contentReference[oaicite:0]{index=0}

What Are Rational Coefficients?

A coefficient is the number multiplying a variable.

A rational number can be written in the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\neq0\).

The Main Strategy

For an equation of the form

\(ax+b=cx+d\)

we want to collect all the \(x\)-terms on one side and all constant terms on the other side.

This follows the textbook’s general sequence: expand and collect like terms, remove the unknown from one side if necessary, and then use inverse operations to isolate it. :contentReference[oaicite:1]{index=1}

Variables on Both Sides

Example: Solve \(5x+7=2x+25\).

We chose to subtract \(2x\) from both sides so that the variable remained only on the left.

 Choosing Which Variable Term to Move

Consider:

\(8x+3=5x+21\)

There are two valid approaches.

Method 1 — Move \(5x\) to the left:

Method 2 — Move \(8x\) to the right:

Both methods produce the same solution.

 Equations with Fractional Coefficients

Fractions can make equations look more complicated than they actually are. There are two main approaches:

• Method 1: Work directly with the fractions.
• Method 2: Multiply every term by the least common denominator (LCD) first.

Example: Solve

\(\dfrac{x}{3}+2=\frac{x}{6}+7\)

Method 1 — Work directly with fractions

Method 2 — Clear the fractions first

The denominators are \(3\) and \(6\), so the LCD is \(6\).

Equations with Several Fractions

When an equation contains several fractional coefficients, clearing the denominators first is often the most efficient method.

Example: Solve

\(\dfrac{2x}{3}-\frac{1}{2}=\frac{x}{6}+5\)

The denominators are \(3\), \(2\), and \(6\), so the LCD is \(6\).

The important point is that the \(6\) must multiply every term, not just the fractions.

Decimal Coefficients

Terminating decimals can also be rational coefficients.

Example: Solve

\(0.8x+3=0.3x+13\)

Alternatively, multiply the entire equation by \(10\):

 Equations with Brackets and Variables on Both Sides

If brackets are present, expand them before collecting the variable terms.

Example: Solve

\(3(x+2)=2x+14\)

For more complicated equations, the same sequence applies: expand → collect → move variable terms → isolate → check. The textbook uses this exact progression when solving equations involving repeated unknowns. :contentReference[oaicite:2]{index=2}

 Three Possible Types of Solutions

An equation with variables on both sides does not necessarily have exactly one solution. It may have:

Example — No solution:

Since \(7=12\) is false, there is no solution.

Example — Infinitely many solutions:

Since the resulting statement is always true, there are infinitely many solutions.

 Checking Solutions

For equations with variables on both sides, substitution into the original equation is an excellent way to verify the result.

For example, for \(5x+7=2x+25\), we found \(x=6\).

Therefore, LHS \(=\) RHS and \(x=6\) is verified.

Common Mistakes