Absolutely! Let's solve each equation step by step, justifying every step along the way.
### Equation 1: [tex]\(10 = 7 - m\)[/tex]
Step 1: Isolate [tex]\(m\)[/tex] by subtracting 7 from both sides.
[tex]\[ 10 - 7 = -m \][/tex]
Justification: We subtract 7 from both sides to move the constant term [tex]\(7\)[/tex] from the right to the left side.
Step 2: Simplify the left side.
[tex]\[ 3 = -m \][/tex]
Justification: Perform the subtraction to simplify the equation.
Step 3: Solve for [tex]\(m\)[/tex] by multiplying both sides by [tex]\(-1\)[/tex].
[tex]\[ m = -3 \][/tex]
Justification: Multiplying by [tex]\(-1\)[/tex] gives the positive value of [tex]\(m\)[/tex].
### Solution for Equation 1: [tex]\( m = -3 \)[/tex]
### Equation 2: [tex]\(\frac{h + 6}{5} = 2\)[/tex]
Step 1: Clear the fraction by multiplying both sides by 5.
[tex]\[ h + 6 = 10 \][/tex]
Justification: Multiplying by the denominator ([tex]\(5\)[/tex]) eliminates the fraction.
Step 2: Isolate [tex]\(h\)[/tex] by subtracting 6 from both sides.
[tex]\[ h = 4 \][/tex]
Justification: Subtracting 6 from both sides moves the constant term from the left to the right side.
### Solution for Equation 2: [tex]\( h = 4 \)[/tex]
### Equation 3: [tex]\(36 = 13n - 4n\)[/tex]
Step 1: Combine like terms on the right side.
[tex]\[ 36 = 9n \][/tex]
Justification: By combining [tex]\(13n\)[/tex] and [tex]\(-4n\)[/tex], we simplify the equation.
Step 2: Solve for [tex]\(n\)[/tex] by dividing both sides by 9.
[tex]\[ n = 4 \][/tex]
Justification: Dividing by 9 isolates the variable [tex]\(n\)[/tex].
### Solution for Equation 3: [tex]\( n = 4 \)[/tex]
### Equation 4: [tex]\(-2(4g - 3) = 30\)[/tex]
Step 1: Distribute [tex]\(-2\)[/tex] on the left side.
[tex]\[ -8g + 6 = 30 \][/tex]
Justification: Using the distributive property, we multiply [tex]\(-2\)[/tex] with each term inside the parentheses.
Step 2: Move the constant term to the right side by subtracting 6 from both sides.
[tex]\[ -8g = 24 \][/tex]
Justification: Subtracting 6 from both sides simplifies the equation.
Step 3: Solve for [tex]\(g\)[/tex] by dividing both sides by [tex]\(-8\)[/tex].
[tex]\[ g = -3 \][/tex]
Justification: Dividing by [tex]\(-8\)[/tex] isolates [tex]\(g\)[/tex].
### Solution for Equation 4: [tex]\( g = -3 \)[/tex]
### Summary:
- [tex]\( m = -3 \)[/tex]
- [tex]\( h = 4 \)[/tex]
- [tex]\( n = 4 \)[/tex]
- [tex]\( g = -3 \)[/tex]
These solutions are derived by systematically applying algebraic principles to isolate the variables in each equation.
