Problem Analysis:
The task involves solving a series of equations and determining whether the provided solutions satisfy the original equations. The goal is to verify each solution step-by-step.
Given Equations and Solutions:
1.
Equation: \( 2x + 3 = 8 \)
-
Solution: \( x = 5 \)
2.
Equation: \( 3x + (24 - x) = 6 \)
-
Solution: \( x = 6 \)
3.
Equation: \( 7y + (9y + 10) = 5(y + 3) \)
-
Solution: \( y = 7 \)
4.
Equation: \( 5(x + 3) + 3 = 6(x + 3) \)
-
Solution: \( x = 2 \)
5.
Equation: \( 3(1 + x) - 2 = 5(x + 3) \)
-
Solution: \( x = -7 \)
Solution Verification:
####
Step 1: Solve Equation 1
-
Equation: \( 2x + 3 = 8 \)
-
Solution: \( x = 5 \)
Substitute \( x = 5 \) into the equation:
\[
2(5) + 3 = 8
\]
\[
10 + 3 = 8
\]
\[
13 \neq 8
\]
Conclusion: The solution \( x = 5 \) does not satisfy the equation.
---
####
Step 2: Solve Equation 2
-
Equation: \( 3x + (24 - x) = 6 \)
-
Solution: \( x = 6 \)
Substitute \( x = 6 \) into the equation:
\[
3(6) + (24 - 6) = 6
\]
\[
18 + 18 = 6
\]
\[
36 \neq 6
\]
Conclusion: The solution \( x = 6 \) does not satisfy the equation.
---
####
Step 3: Solve Equation 3
-
Equation: \( 7y + (9y + 10) = 5(y + 3) \)
-
Solution: \( y = 7 \)
Substitute \( y = 7 \) into the equation:
\[
7(7) + (9(7) + 10) = 5(7 + 3)
\]
\[
49 + (63 + 10) = 5(10)
\]
\[
49 + 73 = 50
\]
\[
122 \neq 50
\]
Conclusion: The solution \( y = 7 \) does not satisfy the equation.
---
####
Step 4: Solve Equation 4
-
Equation: \( 5(x + 3) + 3 = 6(x + 3) \)
-
Solution: \( x = 2 \)
Substitute \( x = 2 \) into the equation:
\[
5(2 + 3) + 3 = 6(2 + 3)
\]
\[
5(5) + 3 = 6(5)
\]
\[
25 + 3 = 30
\]
\[
28 \neq 30
\]
Conclusion: The solution \( x = 2 \) does not satisfy the equation.
---
####
Step 5: Solve Equation 5
-
Equation: \( 3(1 + x) - 2 = 5(x + 3) \)
-
Solution: \( x = -7 \)
Substitute \( x = -7 \) into the equation:
\[
3(1 + (-7)) - 2 = 5((-7) + 3)
\]
\[
3(1 - 7) - 2 = 5(-4)
\]
\[
3(-6) - 2 = -20
\]
\[
-18 - 2 = -20
\]
\[
-20 = -20
\]
Conclusion: The solution \( x = -7 \) satisfies the equation.
---
Final Answer:
\[
\boxed{x = -7}
\]
Parent Tip: Review the logic above to help your child master the concept of linear equation in one variable worksheet.
