To solve the linear equations provided in the worksheet, we will go through each section step by step. Here's a detailed explanation for solving one equation from each section as an example.
---
Section A
####
Problem 1:
\[
\frac{2x + 5}{3} = 11
\]
Solution:
1. Eliminate the denominator by multiplying both sides by 3:
\[
2x + 5 = 33
\]
2. Isolate the term with \( x \) by subtracting 5 from both sides:
\[
2x = 28
\]
3. Solve for \( x \) by dividing both sides by 2:
\[
x = 14
\]
Answer:
\[
\boxed{14}
\]
---
Section B
####
Problem 1:
\[
4(2x - 3) = 8(2x + 5)
\]
Solution:
1. Distribute the constants on both sides:
\[
8x - 12 = 16x + 40
\]
2. Move all terms involving \( x \) to one side and constant terms to the other side:
\[
8x - 16x = 40 + 12
\]
\[
-8x = 52
\]
3. Solve for \( x \) by dividing both sides by -8:
\[
x = -\frac{52}{8} = -\frac{13}{2}
\]
Answer:
\[
\boxed{-\frac{13}{2}}
\]
---
Section C
####
Problem 1:
\[
\frac{5x - 2}{3} = \frac{4x + 1}{2}
\]
Solution:
1. Eliminate the denominators by cross-multiplying:
\[
2(5x - 2) = 3(4x + 1)
\]
2. Distribute the constants:
\[
10x - 4 = 12x + 3
\]
3. Move all terms involving \( x \) to one side and constant terms to the other side:
\[
10x - 12x = 3 + 4
\]
\[
-2x = 7
\]
4. Solve for \( x \) by dividing both sides by -2:
\[
x = -\frac{7}{2}
\]
Answer:
\[
\boxed{-\frac{7}{2}}
\]
---
General Approach for Solving Linear Equations
1.
Eliminate fractions or parentheses by finding a common denominator or distributing.
2.
Simplify both sides by combining like terms.
3.
Isolate the variable by moving all terms involving the variable to one side and constants to the other side.
4.
Solve for the variable by dividing or multiplying as needed.
5.
Check your solution by substituting it back into the original equation.
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If you need solutions for specific problems or further clarification, feel free to ask!
Parent Tip: Review the logic above to help your child master the concept of solving algebraic equations with fractions worksheet.
