Sure! Below is a detailed, step-by-step solution with justifications for each step:
1. Initial equation given:
[tex]\[
HJ = HI + IJ
\][/tex]
Justification: Segment Addition Postulate
- The Segment Addition Postulate states that if [tex]\(HJ\)[/tex] is a segment formed by [tex]\(HI\)[/tex] and [tex]\(IJ\)[/tex], the length of [tex]\(HJ\)[/tex] is the sum of the lengths of [tex]\(HI\)[/tex] and [tex]\(IJ\)[/tex].
2. Substitute the given expressions:
[tex]\[
7x - 3 = (2x + 6) + (3x - 3)
\][/tex]
Justification: Substitution Property of Equality
- We substitute [tex]\(HJ\)[/tex] with [tex]\(7x-3\)[/tex], [tex]\(HI\)[/tex] with [tex]\(2x+6\)[/tex], and [tex]\(IJ\)[/tex] with [tex]\(3x-3\)[/tex] as provided in the problem.
3. Combine like terms on the right-hand side:
[tex]\[
7x - 3 = 5x + 3
\][/tex]
Justification: Simplify
- By combining the like terms on the right-hand side, [tex]\((2x + 6) + (3x - 3)\)[/tex] simplifies to [tex]\(5x + 3\)[/tex].
4. Subtract [tex]\(5x\)[/tex] from both sides:
[tex]\[
7x - 5x - 3 = 3
\][/tex]
which simplifies to:
[tex]\[
2x - 3 = 3
\][/tex]
Justification: Subtraction Property of Equality
- To isolate terms involving [tex]\(x\)[/tex] on one side, subtract [tex]\(5x\)[/tex] from both sides.
5. Add [tex]\(3\)[/tex] to both sides:
[tex]\[
2x - 3 + 3 = 3 + 3
\][/tex]
which simplifies to:
[tex]\[
2x = 6
\][/tex]
Justification: Addition Property of Equality
- To isolate [tex]\(2x\)[/tex], add [tex]\(3\)[/tex] to both sides to cancel out the [tex]\(-3\)[/tex].
6. Divide both sides by [tex]\(2\)[/tex]:
[tex]\[
\frac{2x}{2} = \frac{6}{2}
\][/tex]
which simplifies to:
[tex]\[
x = 3
\][/tex]
Justification: Division Property of Equality
- To solve for [tex]\(x\)[/tex], divide both sides by [tex]\(2\)[/tex] to get [tex]\(x\)[/tex] by itself.
In summary, the systematic application of the Segment Addition Postulate, simplification, and various properties of equality leads to the final solution where [tex]\(x = 3\)[/tex].
