Solve Linear Equations with Parentheses Using the Distributive Law
This tutorial presents detailed examples and exercises on solving linear
equations with parentheses using the distributive property.
Solutions include step-by-step reasoning with explanatory notes.
To remove parentheses, apply the distributive law:
\[
a(b + c) = ab + ac
\]
Examples with Solutions
Example 1
Solve the equation:
\[
-2(x + 3) = x + 6
\]
Solution to Example 1
- Given: \( -2(x+3)=x+6 \)
- Use the distributive law:
\[
-2x - 6 = x + 6
\]
- Add 6 to both sides:
\[
-2x - 6 + 6 = x + 6 + 6
\]
\[
-2x = x + 12
\]
- Subtract \(x\) from both sides:
\[
-2x - x = x + 12 - x
\]
\[
-3x = 12
\]
- Divide both sides by \(-3\):
\[
x = -4
\]
- Check:
Left side: \( -2(-4+3) = 2 \)
Right side: \( -4 + 6 = 2 \)
- Conclusion: \( x = -4 \)
Matched Exercise 1
\[
-3(-x+3)=x-7
\]
Example 2
Solve the equation:
\[
-3(-x - 6) = 3x - 23
\]
Solution to Example 2
- Given:
\[
-3(-x - 6)=3x - 23
\]
- Apply the distributive law:
\[
3x + 18 = 3x - 23
\]
- Subtract 18:
\[
3x = 3x - 41
\]
- Subtract \(3x\) from both sides:
\[
0 = -41
\]
- Conclusion:
No real number satisfies the equation.
The equation has no solution.
Matched Exercise 2
\[
4(-x + 3) = -4x - 7
\]
Example 3
Solve the equation:
\[
-7(x - 6) - 3x - 3 = 3(x + 5) - 2x
\]
Solution to Example 3
- Expand all parentheses:
\[
-7x + 42 - 3x - 3 = 3x + 15 - 2x
\]
- Group like terms:
\[
-10x + 39 = x + 15
\]
- Subtract 39:
\[
-10x = x - 24
\]
- Subtract \(x\):
\[
-11x = -24
\]
- Divide by \(-11\):
\[
x = \frac{24}{11}
\]
- Check:
Both sides evaluate to \( \frac{189}{11} \).
- Conclusion:
\( x = \frac{24}{11} \)
Matched Exercise 3
\[
-5(x - 4) - x + 23 = 5(x - 5) - x
\]
Example 4
Solve the equation:
\[
-\,\frac{2(x - 6)}{7} - \frac{x - 3}{2} = -x
\]
Solution to Example 4
- Multiply both sides by the LCD \(14\):
\[
14\left[-\frac{2(x - 6)}{7} - \frac{x - 3}{2}\right] = 14(-x)
\]
- Simplify:
\[
-4(x-6) - 7(x-3) = -14x
\]
- Expand and combine terms:
\[
-11x + 45 = -14x
\]
- Subtract 45:
\[
-11x = -14x - 45
\]
- Add \(14x\):
\[
3x = -45
\]
- Divide by 3:
\[
x = -15
\]
- Check:
Both sides equal 15.
- Conclusion:
\(x = -15\)
Matched Exercise 4
\[
-3\,\frac{(x + 4)}{4} - x - 2 = \frac{x - 4}{3} - x
\]
More References
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