Let's solve each part step-by-step to determine whether the given [tex]\( x \)[/tex] values are correct solutions and find correct solutions where necessary.
### Part (a)
Equation: [tex]\( 3x + 1 = 16 \)[/tex]\
Given: [tex]\( x = 5 \)[/tex]
1. Substitute [tex]\( x = 5 \)[/tex] into the equation:
[tex]\[ 3(5) + 1 = 15 + 1 = 16 \][/tex]
Since [tex]\( 16 = 16 \)[/tex] holds true, [tex]\( x = 5 \)[/tex] is indeed a solution to the equation [tex]\( 3x + 1 = 16 \)[/tex].
### Part (b)
Equation: [tex]\( 7x = 91 \)[/tex]
Given: [tex]\( x = 13 \)[/tex]
1. Substitute [tex]\( x = 13 \)[/tex] into the equation:
[tex]\[ 7(13) = 91 \][/tex]
Since [tex]\( 91 = 91 \)[/tex] holds true, [tex]\( x = 13 \)[/tex] is indeed a solution to the equation [tex]\( 7x = 91 \)[/tex].
### Part (c)
Equation: [tex]\( 10x + 9 = 7x + 30 \)[/tex]
Given: [tex]\( x = 6 \)[/tex]
1. Substitute [tex]\( x = 6 \)[/tex] into the equation:
[tex]\[ 10(6) + 9 = 60 + 9 = 69 \][/tex]
[tex]\[ 7(6) + 30 = 42 + 30 = 72 \][/tex]
Since [tex]\( 69 \neq 72 \)[/tex], [tex]\( x = 6 \)[/tex] is not a solution.
2. To find the correct solution, let's solve the equation:
[tex]\[ 10x + 9 = 7x + 30 \][/tex]
Subtract [tex]\( 7x \)[/tex] and 9 from both sides:
[tex]\[ 3x = 21 \][/tex]
Divide both sides by 3:
[tex]\[ x = 7 \][/tex]
Therefore, the correct solution to the equation [tex]\( 10x + 9 = 7x + 30 \)[/tex] is [tex]\( x = 7 \)[/tex].
### Part (d)
Equation: [tex]\( -10x - 1 = 29 \)[/tex]
Given: [tex]\( x = 3 \)[/tex]
1. Substitute [tex]\( x = 3 \)[/tex] into the equation:
[tex]\[ -10(3) - 1 = -30 - 1 = -31 \][/tex]
Since [tex]\( -31 \neq 29 \)[/tex], [tex]\( x = 3 \)[/tex] is not a solution.
2. To find the correct solution, let's solve the equation:
[tex]\[ -10x - 1 = 29 \][/tex]
Add 1 to both sides:
[tex]\[ -10x = 30 \][/tex]
Divide both sides by -10:
[tex]\[ x = -3 \][/tex]
Therefore, the correct solution to the equation [tex]\( -10x - 1 = 29 \)[/tex] is [tex]\( x = -3 \)[/tex].
### Summary
- [tex]\( 3x + 1 = 16 \)[/tex] with [tex]\( x = 5 \)[/tex] is correct.
- [tex]\( 7x = 91 \)[/tex] with [tex]\( x = 13 \)[/tex] is correct.
- [tex]\( 10x + 9 = 7x + 30 \)[/tex] with [tex]\( x = 6 \)[/tex] is incorrect. The correct solution is [tex]\( x = 7 \)[/tex].
- [tex]\( -10x - 1 = 29 \)[/tex] with [tex]\( x = 3 \)[/tex] is incorrect. The correct solution is [tex]\( x = -3 \)[/tex].
