Linear Equations
Before we begin our discussion of solving linear equations, let’s review a few definitions.
The procedure used to solve a linear equation varies depending on the equation’s complexity. Is there a variable on just one side of the equation? Is there a variable on both sides of the equation? Is the variable inside a set of parentheses? Are there fractions? The steps below will provide some guidelines in solving equations, and the examples that follow will get into more specifics.
Guide to Solve Equations
The goal is to get the variable isolated on one side of the equation.
NOTE: There will be certain equations that may have no solution or infinite solutions. If you are solving and you get a true statement (like \(10 = 10\) or \(0 = 0\)) then the solution is the set of all real numbers that satisfy the equation. If you get a false statement (like \(10 = 0\)) then there is no real number x that will satisfy the equation, resulting in no solutions.
Let’s look at some examples.
Example 1: Solve the linear equation \(\frac{2}{3}\left(x-4\right)+1=7\).
Before we begin our discussion of solving linear equations, let’s review a few definitions.
- An equation is a statement that two expressions are equal. For example, \(2+1=3\) or \(x+4=6\).
- A linear equation is of the form \(ax+b=c\), where \(a\), \(b\) and \(c\) are constants and \(a\ne0\). For example, \(5x-7=0\).
- If a number is substituted into the equation’s variable, resulting in a true statement, then that number is a solution of the equation.
- Equivalent equations will have the same solution.
The procedure used to solve a linear equation varies depending on the equation’s complexity. Is there a variable on just one side of the equation? Is there a variable on both sides of the equation? Is the variable inside a set of parentheses? Are there fractions? The steps below will provide some guidelines in solving equations, and the examples that follow will get into more specifics.
Guide to Solve Equations
The goal is to get the variable isolated on one side of the equation.
- Perform the distributive property, if necessary.
- Combine like terms on each side of the equation, if necessary.
- Multiply each term by the least common denominator to eliminate fractions, if necessary.
- Use inverse operations (addition or subtraction) to move the variables to the same side of the equal sign, if necessary.
- Use inverse operations (addition or subtraction) to move the constants to the opposite side of the equal sign, if necessary.
- Use inverse operations (division or multiplication by the reciprocal) to eliminate the coefficient of the variable, if necessary.
- Check your solution in the original equation to verify that your solution will provide a true statement.
NOTE: There will be certain equations that may have no solution or infinite solutions. If you are solving and you get a true statement (like \(10 = 10\) or \(0 = 0\)) then the solution is the set of all real numbers that satisfy the equation. If you get a false statement (like \(10 = 0\)) then there is no real number x that will satisfy the equation, resulting in no solutions.
Let’s look at some examples.
Example 1: Solve the linear equation \(\frac{2}{3}\left(x-4\right)+1=7\).
\(\begin{align}\frac{2}{3}\left(x-4\right)+1&=7 \ & \ &\text{1) Write the original equation.}\\\frac{2}{3}\left(x-4\right)&=6\ &\ &\text{2) Subtract 1 from both sides.}\\\frac{2}{3}x-\frac{8}{3}&=6\ &\ &\text{3) Distribute the fraction inside the parentheses.}\\3\left(\frac{2}{3}x-\frac{8}{3}\right)&=3\left(6\right)\ &\ &\text{4) Multiply each term by the least common denominator 3.}\\2x-8&=18\ & \ &\text{ }\\2x&=26\ & \ &\text{5) Add 8 to both sides.}\\x&=13\ & \ &\text{6) Divide both sides by 2.}\end{align}\)
Note: there are other ways you can solve this problem. Many people find it more efficient to eliminate any fractions early in the solving process.
Example 2: Solve the linear equation \(4\left(5-x\right)=2\left(3+2x\right)-5x\).
\(\begin{align}4\left(5-x\right)&=2\left(3+2x\right)-5x \ & \ &\text{1) Rewrite the equation.}\\20-4x&=6+4x-5x \ & \ &\text{2) Distributive Property.}\\20-4x&=6-x \ & \ &\text{3) Combine like terms.}\\20&=6+3x \ & \ &\text{4) Add 4x to both sides.}\\14&=3x \ & \ &\text{5) Subtract 6 from both sides.}\\\frac{14}{3}&=x \ & \ &\text{6) Divide both sides by 3.}\end{align}\)
Example 3: Solve the linear equation \(\frac{1}{2}x+4=-\frac{2}{3}x+\frac{1}{2}\).
Example 4: Solve the linear equation \(2\left(3x+4\right)=5-6\left(2+x\right)\).
Example 5: The bill to repair your car cost \(\$348\). The parts cost \(\$220\) and the labor charge was \(\$32\) per hour. How many hours were spent repairing your car? Set up a linear equation and solve it.