How to Solve 3 Linear Equations in One Variable

This lesson solves three linear equations in one variable. Each one has fractions or decimals, so the plan is the same every time: clear the denominators, expand carefully, then use cross multiplication to finish.

Question 1: an equation with fractions

We are given

$\frac{3x-2}{4} - \frac{2x+3}{3} = \frac{2-3x}{3}.$

Left side. The lowest common denominator of $4$ and $3$ is $12$. Combining the two fractions over $12$:

$\frac{3(3x-2) - 4(2x+3)}{12}.$

Expand the numerator, watching the sign on the second bracket:

$\frac{9x - 6 - 8x - 12}{12} = \frac{x - 18}{12}.$

Cross multiplication. With a single fraction on each side, multiply each numerator by the other side's denominator:

$3(x - 18) = 12(2 - 3x).$

$3x - 54 = 24 - 36x.$

Collect the $x$ terms on the left and the numbers on the right:

$3x + 36x = 24 + 54 \quad\Longrightarrow\quad 39x = 78.$

$x = \frac{78}{39} = 2.$

Question 2: brackets and signs

We are given

$m - \frac{m-1}{2} = 1 - \frac{m-2}{3}.$

Left side. The LCD is $2$:

$\frac{2m - (m-1)}{2} = \frac{2m - m + 1}{2} = \frac{m + 1}{2}.$

Right side. The LCD is $3$:

$\frac{3 - (m-2)}{3} = \frac{3 - m + 2}{3} = \frac{5 - m}{3}.$

Cross multiplication.

$3(m + 1) = 2(5 - m).$

$3m + 3 = 10 - 2m.$

Transpose: move $-2m$ to the left and $+3$ to the right:

$3m + 2m = 10 - 3 \quad\Longrightarrow\quad 5m = 7.$

$m = \frac{7}{5}.$

Question 3: decimal coefficients

We are given

$0.25(4f - 3) = 0.05(10f - 9).$

Write the decimals as fractions: $0.25 = \tfrac{25}{100}$ and $0.05 = \tfrac{5}{100}$. Since both sides share the denominator $100$, it cancels, leaving

$25(4f - 3) = 5(10f - 9).$

Expand both sides:

$100f - 75 = 50f - 45.$

Collect the $f$ terms on the left and the numbers on the right:

$100f - 50f = -45 + 75 \quad\Longrightarrow\quad 50f = 30.$

$f = \frac{30}{50} = \frac{3}{5} = 0.6.$

Key takeaways

• Clear fractions first by multiplying through by the lowest common denominator, and keep each group in brackets.
• When a minus sign sits in front of a bracket, it changes the sign of every term inside.
• With one fraction on each side, cross multiply, then transpose and simplify to isolate the variable. Decimal coefficients can be rewritten as fractions to make this easier.