Example

Solve for [latex]a[/latex]. [latex]4\left(2a+3\right)=28[/latex]

Answer: Apply the distributive property to expand [latex]4\left(2a+3\right)[/latex] to [latex]8a+12[/latex]

[latex]\begin{array}{r}4\left(2a+3\right)=28\\ 8a+12=28\end{array}[/latex]

Subtract [latex]12[/latex] from both sides to isolate the variable term.

[latex]\begin{array}{r}8a+12\,\,\,=\,\,\,28\\ \underline{-12\,\,\,\,\,\,-12}\\ 8a\,\,\,=\,\,\,16\end{array}[/latex]

Divide both terms by [latex]8[/latex] to get a coefficient of [latex]1[/latex].

[latex]\begin{array}{r}\underline{8a}=\underline{16}\\8\,\,\,\,\,\,\,\,\,\,\,\,8\\a\,=\,\,2\end{array}[/latex]

Answer

[latex]a=2[/latex]

Example

Solve for [latex]t[/latex].  [latex]2\left(4t-5\right)=-3\left(2t+1\right)[/latex]

Answer: Apply the distributive property to expand [latex]2\left(4t-5\right)[/latex] to [latex]8t-10[/latex] and [latex]-3\left(2t+1\right)[/latex] to[latex]-6t-3[/latex]. Be careful in this step—you are distributing a negative number, so keep track of the sign of each number after you multiply.

[latex]\begin{array}{r}2\left(4t-5\right)=-3\left(2t+1\right)\,\,\,\,\,\, \\ 8t-10=-6t-3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Add [latex]-6t[/latex] to both sides to begin combining like terms.

[latex]\begin{array}{r}8t-10=-6t-3\\ \underline{+6t\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+6t}\,\,\,\,\,\,\,\\ 14t-10=\,\,\,\,-3\,\,\,\,\,\,\,\end{array}[/latex]

Add 10 to both sides of the equation to isolate t.

[latex]\begin{array}{r}14t-10=-3\\ \underline{+10\,\,\,+10}\\ 14t=\,\,\,7\,\end{array}[/latex]

The last step is to divide both sides by 14 to completely isolate t.

[latex]\begin{array}{r}14t=7\,\,\,\,\\\Large\frac{14t}{14}\normalsize=\Large\frac{7}{14}\end{array}[/latex]

Answer

[latex-display]t=\Large\frac{1}{2}[/latex-display] We simplified the fraction [latex]\Large\frac{7}{14}[/latex] into [latex]\Large\frac{1}{2}[/latex]

Example

Solve  [latex]\Large\frac{1}{2}\normalsize x-3=2-\Large\frac{3}{4}\normalsize x[/latex] by clearing the fractions in the equation first.

Answer: Multiply both sides of the equation by [latex]4[/latex], the common denominator of the fractional coefficients.

[latex]\begin{array}{r}\Large\frac{1}{2}\normalsize{x-3=2-}\Large\frac{3}{4}\normalsize{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\\\\ 4\left(\Large\frac{1}{2}\normalsize{x-3}\right)=4\left(2-\Large\frac{3}{4}\normalsize{x}\right)\end{array}[/latex]

Use the distributive property to expand the expressions on both sides. Multiply.

[latex]\begin{array}{r}4\left(\Large\frac{1}{2}\normalsize{x}\right)-4\left(3\right)=4\left(2\right)-4\left(-\Large\frac{3}{4}\normalsize{x}\right)\\\\ \Large\frac{4}{2}\normalsize{x}-12=8-\Large\frac{12}{4}\normalsize{x}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\\\ 2x-12=8-3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{array}[/latex]

Add 3x to both sides to move the variable terms to only one side.

[latex]\begin{array}{r}2x-12=8-3x\, \\\underline{+3x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+3x}\\ 5x-12=8\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Add 12 to both sides to move the constant terms to the other side.

[latex]\begin{array}{r}5x-12=8\,\,\\ \underline{\,\,\,\,\,\,+12\,+12} \\5x=20\end{array}[/latex]

Divide to isolate the variable.

[latex]\begin{array}{r}\underline{5x}=\underline{5}\\ 5\,\,\,\,\,\,\,\,\,5\\ x=4\end{array}[/latex]

Answer

[latex]x=4[/latex]

Example

Solve [latex]3y+10.5=6.5+2.5y[/latex] by clearing the decimals in the equation first.

Answer: Since the smallest decimal place represented in the equation is 0.10, we want to multiply by 10 to make 1.0 and clear the decimals from the equation.

[latex]\begin{array}{r}3y+10.5=6.5+2.5y\,\,\,\,\,\,\,\,\,\,\,\,\\\\ 10\left(3y+10.5\right)=10\left(6.5+2.5y\right)\end{array}[/latex]

Use the distributive property to expand the expressions on both sides.

[latex]\begin{array}{r}10\left(3y\right)+10\left(10.5\right)=10\left(6.5\right)+10\left(2.5y\right)\end{array}[/latex]

Multiply.

[latex]30y+105=65+25y[/latex]

Move the smaller variable term, [latex]25y[/latex], by subtracting it from both sides.

[latex]\begin{array}{r}30y+105=65+25y\,\,\\ \underline{-25y\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-25y} \\5y+105=65\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\end{array}[/latex]

Subtract 105 from both sides to isolate the term with the variable.

[latex]\begin{array}{r}5y+105=65\,\,\,\\ \underline{\,\,\,\,\,\,-105\,-105} \\5y=-40\end{array}[/latex]

Divide both sides by 5 to isolate the y.

[latex]\begin{array}{l}\underline{5y}=\underline{-40}\\ 5\,\,\,\,\,\,\,\,\,\,\,\,\,5\\ \,\,\,x=-8\end{array}[/latex]

Answer

[latex]x=-8[/latex]