1.1 Solving Linear Equations
Step 1: Simplify Both Sides of the Equation
Before you start solving for the variable, make sure the equation is simplified as much as possible. This means:
- Combine like terms on each side of the equation.
- Apply the distributive property if necessary (i.e., expand brackets).
Step 2: Move Variable Terms to One Side
The goal is to isolate the variable on one side of the equation. If there are terms with the variable on both sides of the equation, use addition or subtraction to move all variable terms to one side.
Step 3: Move Constant Terms to the Opposite Side
Use addition or subtraction to move all constant terms (numbers without variables) to the opposite side of the equation from the variable.
Step 4: Solve for the Variable
Now, you should have an equation that looks like ax = b, where a and b are numbers. Solve for the variable by dividing both sides of the equation by the coefficient of the variable.
Step 5: Check Your Solution
Substitute your solution back into the original equation to verify that it’s correct.
Example 1: Solve the equation \(2x + 3 = 7x - 1\).
- Simplify Both Sides: The equation is already simplified.
- Move Variable Terms to One Side: Subtract \(2x\) from both sides to get \(3 = 5x - 1\).
- Move Constant Terms to the Opposite Side: Add 1 to both sides to get \(4 = 5x\).
- Solve for the Variable: Divide both sides by 5 to get \(x = frac{4}{5}\).
- Check Your Solution: Substitute \(x = \frac{4}{5}\) into the original equation to verify that it’s correct.
Example 2: Solve the equation \(3x – 2 + 2x = 4x – 1 + 3\).
- Simplify Both Sides: Combine like terms on each side of the equation.
On the left side, \(3x – 2 + 2x\) simplifies to \(5x - 2\).
On the right side, \(4x – 1 + 3\) simplifies to \(4x + 2\). So, the equation becomes \(5x – 2 = 4x + 2\). - Move Variable Terms to One Side: Subtract \(4x\) from both sides to get \(x – 2 = 2\).
- Move Constant Terms to the Opposite Side: Add 2 to both sides to get \(x = 4\).
- Solve for the Variable: In this case, \(x\) is already isolated, so \(x = 4\) is the solution.
- Check Your Solution: Substitute \(x = 4\) into the original equation to verify that it’s correct.
Let’s go through the process of solving linear equations step by step. We’ll start with simple equations and gradually move to more complex ones.
Step 1: Simple Linear Equations
A simple linear equation is of the form \(ax = b\), where a and b are numbers and \(x\) is the variable we want to solve for.
Example: Solve for \(x\) in the equation \(3x = 6\).
To solve for \(x\), we divide both sides of the equation by 3:
\(x=\frac{6}{3}=2\)
So, \(x = 2\) is the solution.
Step 2: Linear Equations with Addition or Subtraction
Sometimes, the equation might have a number added or subtracted from x. In this case, we use addition or subtraction to isolate x.
Example: Solve for \(x\) in the equation \(x - 4 = 7\).
To solve for \(x\), we add 4 to both sides of the equation:
\(x = 7 + 4 = 11\)
So, \(x = 11\) is the solution.
Step 3: Linear Equations with Multiplication and Division
In some cases, \(x\) might be multiplied or divided by a number. Here, we use multiplication or division to solve for \(x\).
Example: Solve for \(x\) in the equation \(5x = 20\).
To solve for \(x\), we divide both sides of the equation by 5:
\(x =\frac{20}{5}= 4\)
So, \(x = 4\) is the solution.
Step 4: Multi-Step Linear Equations
Multi-step linear equations involve more than one operation. To solve these, we use the order of operations (PEMDAS/BODMAS).
Example: Solve for \(x\) in the equation \(2x + 3 = 7\).
First, subtract 3 from both sides:
\(2x = 7 - 3 = 4\)
Then, divide both sides by 2:
\(x =\frac{4}{2}= 2\)
So, \(x = 2\) is the solution.