Easy Way to Solve Linear Equations in One Variable

This lesson is about solving linear equations in one variable. A solution is the value of the variable that makes the equation true, and the process of finding it is called solving the equation.

The transposition rule

When you move a term from one side of the equation to the other, its operation reverses: addition becomes subtraction, subtraction becomes addition, multiplication becomes division, and division becomes multiplication.

For example, $x + 2 = 5$. Transpose $+2$ to the right side, where it becomes $-2$:

$x = 5 - 2 = 3$

Collecting like terms

Solve $2x - 5 = x + 3$. Collect the $x$ terms on the left and the numbers on the right:

$2x - x = 3 + 5$ $x = 8$

Equations with brackets

Solve $3(x + 5) = 2x - 8$. First expand the bracket:

$3x + 15 = 2x - 8$

Keep the terms already on the right where they are, then transpose:

$3x - 2x = -8 - 15$ $x = -23$

Next, solve $3m - 2(m + 4) = 4(m - 2) + 20$. Expand both sides:

$3m - 2m - 8 = 4m - 8 + 20$ $m - 8 = 4m + 12$

Transpose the $m$ terms to the left and the numbers to the right:

$m - 4m = 12 + 8$ $-3m = 20$ $m = -\tfrac{20}{3}$

Equations with equal denominators

Solve $\tfrac{2x + 4}{5} = \tfrac{3x - 1}{5}$. The denominators are the same, so compare the numerators:

$2x - 3x = -1 - 4$ $-x = -5$ $x = 5$

Clearing fractions with the LCM

Solve $\tfrac{x}{2} + 5x + 7 = \tfrac{3x}{2} - 6$. The LCM of the denominators is $2$, so multiply every term by $2$:

$x + 10x + 14 = 3x - 12$ $11x + 14 = 3x - 12$

Now transpose:

$11x - 3x = -12 - 14$ $8x = -26$ $x = -\tfrac{26}{8} = -\tfrac{13}{4}$

Solving by cross multiplication

Solve $\tfrac{2x + 5}{5} = \tfrac{3x - 1}{2}$. Multiply each numerator by the opposite denominator:

$2(2x + 5) = 5(3x - 1)$ $4x + 10 = 15x - 5$ $4x - 15x = -5 - 10$ $-11x = -15$ $x = \tfrac{15}{11}$

Key takeaways

• Transposing a term to the other side reverses its operation.
• Gather the variable on one side and the constants on the other, then simplify.
• Clear fractions by multiplying every term by the LCM, or by cross multiplying when each side is a single fraction.