Certainly! Let's break down Heidi's solution step by step, providing justification for each step:
Step 1:
Equation: [tex]\(3(x+4) + 2 = 2 + 5(x-4)\)[/tex]
Justification: Apply the distributive property
Here, Heidi distributed the multiplication over addition/subtraction in the parentheses:
[tex]\[ 3(x+4) \rightarrow 3x + 12 \][/tex]
[tex]\[ 5(x-4) \rightarrow 5x - 20 \][/tex]
Then she combined the results:
[tex]\[ 3x + 12 + 2 = 2 + 5x - 20 \][/tex]
Step 2:
Equation: [tex]\(3x + 14 = 5x - 18\)[/tex]
Justification: Combine like terms
Heidi combined the constants on each side of the equation:
[tex]\[ 3x + 12 + 2 \rightarrow 3x + 14 \][/tex]
[tex]\[ 2 + 5x - 20 \rightarrow 5x - 18 \][/tex]
Step 3:
Equation: [tex]\(14 = 2x - 18\)[/tex]
Justification: Move all variable terms to one side and constant terms to the other
Heidi subtracted [tex]\(3x\)[/tex] from both sides to move the variable terms to one side:
[tex]\[ 3x + 14 - 3x = 5x - 18 - 3x \][/tex]
Simplifying gave:
[tex]\[ 14 = 2x - 18 \][/tex]
Step 4:
Equation: [tex]\(32 = 2x\)[/tex]
Justification: Isolate the variable term
Heidi added [tex]\(18\)[/tex] to both sides to isolate the term containing [tex]\(x\)[/tex]:
[tex]\[ 14 + 18 = 2x - 18 + 18 \][/tex]
Simplifying gave:
[tex]\[ 32 = 2x \][/tex]
Step 5:
Equation: [tex]\(16 = x\)[/tex]
Justification: Solve for [tex]\(x\)[/tex] by dividing both sides by 2
Heidi divided both sides of the equation by [tex]\(2\)[/tex]:
[tex]\[ \frac{32}{2} = \frac{2x}{2} \][/tex]
Simplifying gave:
[tex]\[ 16 = x \][/tex]
In summary, Heidi solved the equation step-by-step using the distributive property, combining like terms, moving variable and constant terms appropriately, isolating the variable term, and finally solving for [tex]\(x\)[/tex].
