Solving Linear Equations
- Page ID
- 29779
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Solving Linear Equations in One Variable
Linear equations in one variable involve two expressions separated by an \(=\) sign where at least one side of the equation contains the variable \(x\).
Example 1: \(0.4(60-c)=10\)
Example 2: \(\frac{c-16}{18}=0.5\)
Example 3: \(7=\frac{x-6}{\frac{9}{\sqrt{36}}}\)
Example 4: \(0.7(x-1)=28\) Practice on your own. (Solution: \(x=41\))
To solve linear equations perform the following steps. (Note: sometimes, some of the steps may not be necessary.)
Solution to Example 1: \(0.4(60-c)=10\)
Step 1: Eliminate parentheses by multiplying the number outside the parentheses by each number inside the parentheses.
\(0.4(60-c)=10\)
\(0.4\times 60-0.4c = 10\)
\(24-4c = 10\)
Step 2: Eliminate any denominators (if there are any) in the expression containing the variable by multiplying each side of the equation by the denominator.
Note: This is not necessary in this example because \(24-0.4c\) does not contain a denominator.
Step 3: Isolate the variable on one side of the \(=\) sign and the number on the opposite side by adding the opposite to each side of the equation.
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Subtract 24 from each side to get \(-0.4c\) alone on the left side |
\(24-0.4c = 10\) |
Step 4: Isolate the variable by dividing each side of the equation by the number in front of the variable.
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Divide each side by \(-0.4\) |
\(\frac{-0.4c}{-0.4} = \frac{-14}{-0.4}\) |
\(c=35\) answer |
Solution to Example 2: Solve: \(\frac{c-16}{18}=0.5\)
Step 1: Not necessary (there are no parentheses)
Step 2: Multiply each side by 18 to eliminate the denominator:
\(\frac{18}{1}\cdot \frac{c-16}{18} = 0.5 \cdot 18, c - 18 = 9\)
Step 3: Isolate 
by adding 16 to each side
\(c - 16 = 9\)
\(+16\) \(+16\)
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\(c = 25\) answer
Step 4: Not necessary because no number is in front of \(c\).
Solution to Example 3: Solve: \(7=\frac{x-6}{\frac{9}{\sqrt{36}}}\)
Step 1: Not necessary since there are no parentheses.
Step 2: Multiply each side by \(\frac{9}{\sqrt{36}}\) to eliminate the fraction.
\(7\cdot \frac{9}{\sqrt{36}}\cdot \frac{x-6}{\frac{9}{\sqrt{36}}}\)
\(7\cdot \frac{9}{6} = x - 6\)
\(7\cdot 1.5 = x - 6\)
\(10.5 = x - 6\)
Step 3: Add 6 to each side to isolate \(x\).
\(10.5 = x-6\)
\(+6\) = \(+6\)
\(16.5 = x\) answer
Step 4: Not necessary – no number is in front of \(x\).