The following steps will help you solve linear equations.
Example 1
Solve the equation below for \(x\). \[2(3-x)+4x-1=4-2(x+1)-3 \]
Solution
Let's see these steps in action.
Step 1: Expand the brackets.
\[6-2x+4x=4-2x-2-3\]
Step 2: Gather terms:.
\[-2x+4x+2x=4-2-3-6\]
Step 3: Combine like terms.
\[4x = -7\]
Step 4: Divide across to isolate the variable.
\[x = {-7 \over 4}\]
Example 2
Solve the equation below for \(x\). \[8-\frac{7x }{4}=15 \]
Solution
Step 1: Multiply both sides of the equation by the LCM, 4
\[4(8-\frac{7x }{4})=4\times15 \]
Step 2: Remove brackets
\[4\times8 - 4(\frac{7x }{4})=4\times15 \]
Step 3: Simplify
\[32 - 7x = 60\]
Step 4: Collect like terms
\[-7x=60-32\]
Step 5: Combine like terms
\[-7x=28 \]
Step 6: Divide both sides by the coefficient of \(x\)
\[x=-4 \]
Example 3
Solve the equation below for \(x\). \[3(x-m)=12-x\]
Solution
Step 1: Remove brackets
\[3x-3m=12-x \]
Step 2: Collect like terms
\[3x+x=12+3m\]
Step 3: Combine like terms
\[4x=12+3m \]
Step 4: Divide both sides by the coefficient of \(x\)
\[x=3+ \frac{3m}{4} \]
Example 4
Solve the equation below for \(x\). \[\frac{5(x-1)}{6} - \frac{3x+11}{8}=1\]
Solution
Step 1: Multiply both sides of the equation by the LCM, 48
\[48\times\frac{5(x-1)}{6} - 48\times\frac{3x+11}{8}=1\times48\]
Step 2: Simplify
\[8\times5(x-1) - 6(3x+11)=48 \]
Step 3: Expand the brackets
\[40x-40 - 18x-66=48 \]
Step 4: Collect like terms
\[40x-18x=48+40+66\]
Step 5: Combine like terms
\[22x=154 \]
Step 6: Divide both sides by the coefficient of \(x\)
\[x=7 \]