Equations with Unknowns on Both Sides

An equation is a mathematical statement that shows two expressions are equal, often containing unknown values called variables. In some equations, the variable appears on both sides of the equals sign; these are called equations with unknowns on both sides.

Identify Equations with Unknowns Both Sides

Equations with unknowns on both sides contain the variable (usually x) on each side of the equals sign. Recognising these is the first step to solving them.

Examples include:

• $3x + 5 = 2x + 10$
• $7y - 4 = 3y + 12$
• $5a + 2 = 5a - 3$

To solve such equations, you first set up the equation clearly, ensuring both sides are written out fully. The goal is to isolate the variable on one side.

For instance, in the equation $3x + 5 = 2x + 10$, the variable x appears on both sides. We will rearrange terms to get all x terms on one side and constants on the other.

Isolate Variable Terms

To isolate the variable terms on one side, you add or subtract terms containing the variable on both sides of the equation. Remember, whatever you do to one side, you must do to the other to keep the equation balanced.

Steps to isolate variable terms:

• Choose one side to collect all variable terms.
• Add or subtract terms containing the variable on both sides to move them.
• Do the same with constant terms to move them to the opposite side.
• Simplify both sides by combining like terms (numbers and variables separately).

For example, starting with $3x + 5 = 2x + 10$, subtract $2x$ from both sides:

$3x + 5 - 2x = 2x + 10 - 2x$

Simplifies to:

$x + 5 = 10$

Next, subtract 5 from both sides:

$x + 5 - 5 = 10 - 5$

Which gives:

$x = 5$

Solve Simplified Equation

Once the variable terms are isolated on one side and constants on the other, solve the equation by dividing or multiplying to find the value of the variable.

Check for special cases:

• If the variable terms cancel out and constants are equal, there are infinitely many solutions.
• If the variable terms cancel out but constants are not equal, there is no solution.

Always verify your solution by substituting it back into the original equation to ensure both sides are equal.

For example, if you find $x = 5$, substitute back into $3x + 5 = 2x + 10$:

$3(5) + 5 = 15 + 5 = 20$

$2(5) + 10 = 10 + 10 = 20$

Both sides equal 20, so $x = 5$ is correct.