Solving Linear Equations
When we talk about linear equations, we're often referring to equations that can be graphically represented as straight lines on a coordinate plane. In this article, we'll focus on solving simple linear equations with one variable, a fundamental skill in pre-algebra that is essential for understanding more complex algebraic concepts later on.
Understanding Linear Equations
A linear equation in one variable is typically written in the form:
\[ ax + b = c \]
Where:
- \( a \) is a coefficient,
- \( x \) is the variable we want to solve for,
- \( b \) and \( c \) are constants.
The goal in solving the equation is to isolate \( x \) on one side of the equation to determine its value.
Step-by-Step Guide to Solving Linear Equations
Here are the steps to follow when solving a simple linear equation:
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Identify the equation: Look at the form of the equation and identify the coefficients and constants.
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Isolate the variable: Use arithmetic operations (addition, subtraction, multiplication, and division) to move all terms involving the variable to one side and all constant terms to the other side.
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Perform the inverse operations: If \( x \) is being added to \( b \), then subtract \( b \) from both sides. If \( x \) is being multiplied by \( a \), then divide both sides by \( a \).
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Simplify: After performing the necessary operations, simplify the equation to find the value of \( x \).
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Check your work: Substitute your solution back into the original equation to verify its accuracy.
Example Problems
Let's practice with some example problems to illustrate these steps.
Example 1
Solve the equation:
\[ 2x + 3 = 11 \]
Step 1: Isolate the variable
Subtract 3 from both sides:
\[ 2x + 3 - 3 = 11 - 3 \]
\[ 2x = 8 \]
Step 2: Solve for \( x \)
Now divide both sides by 2:
\[ \frac{2x}{2} = \frac{8}{2} \]
\[ x = 4 \]
Checking the solution:
Substitute \( x \) back into the original equation:
\[ 2(4) + 3 = 11 \]
\[ 8 + 3 = 11 \]
This is correct!
Example 2
Now, let’s try another one:
\[ 5x - 7 = 18 \]
Step 1: Isolate the variable
Add 7 to both sides:
\[ 5x - 7 + 7 = 18 + 7 \]
\[ 5x = 25 \]
Step 2: Solve for \( x \)
Divide both sides by 5:
\[ \frac{5x}{5} = \frac{25}{5} \]
\[ x = 5 \]
Checking the solution:
Substitute \( x \) back into the original equation:
\[ 5(5) - 7 = 18 \]
\[ 25 - 7 = 18 \]
This works too!
Example 3
Let’s try solving an equation that incorporates a little more complexity:
\[ 3(x - 2) = 9 \]
Step 1: Distribute the 3
Expand the left side:
\[ 3x - 6 = 9 \]
Step 2: Isolate the variable
Add 6 to both sides:
\[ 3x - 6 + 6 = 9 + 6 \]
\[ 3x = 15 \]
Step 3: Solve for \( x \)
Divide both sides by 3:
\[ \frac{3x}{3} = \frac{15}{3} \]
\[ x = 5 \]
Checking the solution:
Substitute \( x \) back into the original equation:
\[ 3(5 - 2) = 9 \]
\[ 3(3) = 9 \]
It checks out!
Practice Problems
Now it’s your turn! Try solving these equations on your own:
- \( 4x + 1 = 17 \)
- \( -3x + 6 = 0 \)
- \( \frac{1}{2}x - 4 = 2 \)
- \( 6 - 2x = 4 \)
- \( 7(x + 3) = 49 \)
Answers to Practice Problems
Once you've attempted the problems, check your solutions below:
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\( 4x + 1 = 17 \)
Subtract 1: \( 4x = 16 \)
Divide by 4: \( x = 4 \)
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\( -3x + 6 = 0 \)
Subtract 6: \( -3x = -6 \)
Divide by -3: \( x = 2 \)
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\( \frac{1}{2}x - 4 = 2 \)
Add 4: \( \frac{1}{2}x = 6 \)
Multiply by 2: \( x = 12 \)
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\( 6 - 2x = 4 \)
Subtract 6: \( -2x = -2 \)
Divide by -2: \( x = 1 \)
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\( 7(x + 3) = 49 \)
Divide by 7: \( x + 3 = 7 \)
Subtract 3: \( x = 4 \)
Conclusion
Solving linear equations is a critical skill that forms the foundation of algebra. The more you practice, the more proficient you will become. Remember to isolate the variable and check your work by substituting the solution back into the original equation. With these skills in your toolbox, you're now ready to tackle more complex algebraic concepts with confidence! Keep practicing, and don't hesitate to go back to the basics whenever you need a refresher. Happy solving!