Solving equations by multiplication and division involves using inverse operations to isolate a variable. If a variable is multiplied by a number, divide both sides by that number; if it is divided by a number, multiply both sides by that number. These rules work with positive numbers, negative numbers, fractions, and decimals, and they help build the foundation for more advanced algebra.
Lesson
Questions
Solve for x.
\(\textbf{1)}\) \(4x=16\)The answer is \(x=4\)
\(4x=16\)
\(\frac{4x}{4}=\frac{16}{4}\) (Divide both sides by 4.)
\(x=4\)
\(4x=16\)
\(\frac{4x}{4}=\frac{16}{4}\) (Divide both sides by 4.)
\(x=4\)
\(\textbf{2)}\) \(8x=48\)
The answer is \(x=6\)
\(8x=48\)
\(\frac{8x}{8}=\frac{48}{8}\) (Divide both sides by 8.)
\(x=6\)
\(8x=48\)
\(\frac{8x}{8}=\frac{48}{8}\) (Divide both sides by 8.)
\(x=6\)
\(\textbf{3)}\) \(\frac{x}{4}=8\)
The answer is \(x=32\)
\(\frac{x}{4}=8\)
\(4\cdot\frac{x}{4}=8\cdot4\) (Multiply both sides by 4.)
\(x=32\)
\(\frac{x}{4}=8\)
\(4\cdot\frac{x}{4}=8\cdot4\) (Multiply both sides by 4.)
\(x=32\)
\(\textbf{4)}\) \(\frac{x}{3}=-14\)
The answer is \(x=-42\)
\(\frac{x}{3}=-14\)
\(3\cdot\frac{x}{3}=-14\cdot3\) (Multiply both sides by 3.)
\(x=-42\)
\(\frac{x}{3}=-14\)
\(3\cdot\frac{x}{3}=-14\cdot3\) (Multiply both sides by 3.)
\(x=-42\)
\(\textbf{5)}\) \(\frac{x}{2}=3\)
The answer is \(x=6\)
\(\frac{x}{2}=3\)
\(2\cdot\frac{x}{2}=3\cdot2\) (Multiply both sides by 2.)
\(x=6\)
\(\frac{x}{2}=3\)
\(2\cdot\frac{x}{2}=3\cdot2\) (Multiply both sides by 2.)
\(x=6\)
\(\textbf{6)}\) \(9x=3\)
The answer is \(x=\frac{1}{3}\)
\(9x=3\)
\(\frac{9x}{9}=\frac{3}{9}\) (Divide both sides by 9.)
\(x=\frac{3}{9}\)
\(x=\frac{1}{3}\) (Simplify.)
\(9x=3\)
\(\frac{9x}{9}=\frac{3}{9}\) (Divide both sides by 9.)
\(x=\frac{3}{9}\)
\(x=\frac{1}{3}\) (Simplify.)
\(\textbf{7)}\) \(7x=56\)
The answer is \(x=8\)
\(7x=56\)
\(\frac{7x}{7}=\frac{56}{7}\) (Divide both sides by 7.)
\(x=8\)
\(7x=56\)
\(\frac{7x}{7}=\frac{56}{7}\) (Divide both sides by 7.)
\(x=8\)
\(\textbf{8)}\) \(6x=-42\)
The answer is \(x=-7\)
\(6x=-42\)
\(\frac{6x}{6}=\frac{-42}{6}\) (Divide both sides by 6.)
\(x=-7\)
\(6x=-42\)
\(\frac{6x}{6}=\frac{-42}{6}\) (Divide both sides by 6.)
\(x=-7\)
\(\textbf{9)}\) \(-5x=45\)
The answer is \(x=-9\)
\(-5x=45\)
\(\frac{-5x}{-5}=\frac{45}{-5}\) (Divide both sides by -5.)
\(x=-9\)
\(-5x=45\)
\(\frac{-5x}{-5}=\frac{45}{-5}\) (Divide both sides by -5.)
\(x=-9\)
\(\textbf{10)}\) \(-4x=-28\)
The answer is \(x=7\)
\(-4x=-28\)
\(\frac{-4x}{-4}=\frac{-28}{-4}\) (Divide both sides by -4.)
\(x=7\)
\(-4x=-28\)
\(\frac{-4x}{-4}=\frac{-28}{-4}\) (Divide both sides by -4.)
\(x=7\)
\(\textbf{11)}\) \(\frac{x}{5}=-9\)
The answer is \(x=-45\)
\(\frac{x}{5}=-9\)
\(5\cdot\frac{x}{5}=-9\cdot5\) (Multiply both sides by 5.)
\(x=-45\)
\(\frac{x}{5}=-9\)
\(5\cdot\frac{x}{5}=-9\cdot5\) (Multiply both sides by 5.)
\(x=-45\)
\(\textbf{12)}\) \(\frac{x}{-6}=7\)
The answer is \(x=-42\)
\(\frac{x}{-6}=7\)
\(-6\cdot\frac{x}{-6}=7(-6)\) (Multiply both sides by -6.)
\(x=-42\)
\(\frac{x}{-6}=7\)
\(-6\cdot\frac{x}{-6}=7(-6)\) (Multiply both sides by -6.)
\(x=-42\)
\(\textbf{13)}\) \(\frac{x}{-8}=-5\)
The answer is \(x=40\)
\(\frac{x}{-8}=-5\)
\(-8\cdot\frac{x}{-8}=(-5)(-8)\) (Multiply both sides by -8.)
\(x=40\)
\(\frac{x}{-8}=-5\)
\(-8\cdot\frac{x}{-8}=(-5)(-8)\) (Multiply both sides by -8.)
\(x=40\)
\(\textbf{14)}\) \(3x=0\)
The answer is \(x=0\)
\(3x=0\)
\(\frac{3x}{3}=\frac{0}{3}\) (Divide both sides by 3.)
\(x=0\)
\(3x=0\)
\(\frac{3x}{3}=\frac{0}{3}\) (Divide both sides by 3.)
\(x=0\)
\(\textbf{15)}\) \(12x=8\)
The answer is \(x=\frac{2}{3}\)
\(12x=8\)
\(\frac{12x}{12}=\frac{8}{12}\) (Divide both sides by 12.)
\(x=\frac{8}{12}\)
\(x=\frac{2}{3}\) (Simplify.)
\(12x=8\)
\(\frac{12x}{12}=\frac{8}{12}\) (Divide both sides by 12.)
\(x=\frac{8}{12}\)
\(x=\frac{2}{3}\) (Simplify.)
\(\textbf{16)}\) \(\frac{x}{7}=\frac{3}{2}\)
The answer is \(x=\frac{21}{2}\)
\(\frac{x}{7}=\frac{3}{2}\)
\(7\cdot\frac{x}{7}=7\cdot\frac{3}{2}\) (Multiply both sides by 7.)
\(x=\frac{21}{2}\)
\(\frac{x}{7}=\frac{3}{2}\)
\(7\cdot\frac{x}{7}=7\cdot\frac{3}{2}\) (Multiply both sides by 7.)
\(x=\frac{21}{2}\)
\(\textbf{17)}\) \(\frac{2}{3}x=8\)
The answer is \(x=12\)
\(\frac{2}{3}x=8\)
\(\frac{3}{2}\cdot\frac{2}{3}x=8\cdot\frac{3}{2}\) (Multiply both sides by the reciprocal, 3/2.)
\(x=\frac{24}{2}\)
\(x=12\)
\(\frac{2}{3}x=8\)
\(\frac{3}{2}\cdot\frac{2}{3}x=8\cdot\frac{3}{2}\) (Multiply both sides by the reciprocal, 3/2.)
\(x=\frac{24}{2}\)
\(x=12\)
\(\textbf{18)}\) \(\frac{3}{4}x=-9\)
The answer is \(x=-12\)
\(\frac{3}{4}x=-9\)
\(\frac{4}{3}\cdot\frac{3}{4}x=-9\cdot\frac{4}{3}\) (Multiply both sides by 4/3.)
\(x=\frac{-36}{3}\)
\(x=-12\)
The answer is \(x=-12\)
\(\frac{3}{4}x=-9\)
\(\frac{4}{3}\cdot\frac{3}{4}x=-9\cdot\frac{4}{3}\) (Multiply both sides by 4/3.)
\(x=\frac{-36}{3}\)
\(x=-12\)
\(\textbf{19)}\) \(-\frac{5}{2}x=15\)
The answer is \(x=-6\)
\(-\frac{5}{2}x=15\)
\(-\frac{2}{5}\cdot\left(-\frac{5}{2}x\right)=15\left(-\frac{2}{5}\right)\) (Multiply by the reciprocal, -2/5.)
\(x=\frac{-30}{5}\)
\(x=-6\)
\(-\frac{5}{2}x=15\)
\(-\frac{2}{5}\cdot\left(-\frac{5}{2}x\right)=15\left(-\frac{2}{5}\right)\) (Multiply by the reciprocal, -2/5.)
\(x=\frac{-30}{5}\)
\(x=-6\)
\(\textbf{20)}\) \(\frac{x}{\frac{3}{5}}=10\)
The answer is \(x=6\)
\(\frac{x}{\frac{3}{5}}=10\)
\(\frac{3}{5}\cdot\frac{x}{\frac{3}{5}}=10\cdot\frac{3}{5}\) (Multiply both sides by 3/5.)
\(x=\frac{30}{5}\)
\(x=6\)
\(\frac{x}{\frac{3}{5}}=10\)
\(\frac{3}{5}\cdot\frac{x}{\frac{3}{5}}=10\cdot\frac{3}{5}\) (Multiply both sides by 3/5.)
\(x=\frac{30}{5}\)
\(x=6\)
\(\textbf{21)}\) \(0.5x=7\)
The answer is \(x=14\)
\(0.5x=7\)
\(\frac{0.5x}{0.5}=\frac{7}{0.5}\) (Divide both sides by 0.5.)
\(x=14\)
\(0.5x=7\)
\(\frac{0.5x}{0.5}=\frac{7}{0.5}\) (Divide both sides by 0.5.)
\(x=14\)
\(\textbf{22)}\) \(1.2x=-6\)
The answer is \(x=-5\)
\(1.2x=-6\)
\(\frac{1.2x}{1.2}=\frac{-6}{1.2}\) (Divide both sides by 1.2.)
\(x=-5\)
\(1.2x=-6\)
\(\frac{1.2x}{1.2}=\frac{-6}{1.2}\) (Divide both sides by 1.2.)
\(x=-5\)
\(\textbf{23)}\) \(\frac{x}{0.4}=15\)
The answer is \(x=6\)
\(\frac{x}{0.4}=15\)
\(0.4\cdot\frac{x}{0.4}=15(0.4)\) (Multiply both sides by 0.4.)
\(x=6\)
\(\frac{x}{0.4}=15\)
\(0.4\cdot\frac{x}{0.4}=15(0.4)\) (Multiply both sides by 0.4.)
\(x=6\)
\(\textbf{24)}\) \(0.25x=3.5\)
The answer is \(x=14\)
\(0.25x=3.5\)
\(\frac{0.25x}{0.25}=\frac{3.5}{0.25}\) (Divide both sides by 0.25.)
\(x=14\)
\(0.25x=3.5\)
\(\frac{0.25x}{0.25}=\frac{3.5}{0.25}\) (Divide both sides by 0.25.)
\(x=14\)
\(\textbf{25)}\) \(-\frac{x}{2.5}=4\)
The answer is \(x=-10\)
\(-\frac{x}{2.5}=4\)
\(\frac{x}{-2.5}=4\) (Rewrite the negative sign in the denominator.)
\(-2.5\cdot\frac{x}{-2.5}=4(-2.5)\) (Multiply both sides by -2.5.)
\(x=-10\)
The answer is \(x=-10\)
\(-\frac{x}{2.5}=4\)
\(\frac{x}{-2.5}=4\) (Rewrite the negative sign in the denominator.)
\(-2.5\cdot\frac{x}{-2.5}=4(-2.5)\) (Multiply both sides by -2.5.)
\(x=-10\)
See Related Pages
\(\bullet\text{ Equation Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Solving Equations by Addition and Subtraction}\)
\(\,\,\,\,\,\,\,\,x+3=4…\)
\(\bullet\text{ Solving Equations by Multiplication and Division}\)
\(\,\,\,\,\,\,\,\,8x=48…\)
\(\bullet\text{ Solving Multi-step Equations}\)
\(\,\,\,\,\,\,\,\,3x+2=14…\)
\(\bullet\text{ Solving Equations with Variables on Both Sides}\)
\(\,\,\,\,\,\,\,\,3x+5=7x-3…\)
\(\bullet\text{ Solving Equations with Decimals}\)
\(\,\,\,\,\,\,\,\,43.5+0.2x=51.1…\)
\(\bullet\text{ Solving Equations with Fractions}\)
\(\,\,\,\,\,\,\,\,\frac{2}{5}x+\frac{2}{3}=\frac{8}{3}…\)