Question

complete the following proofs.
1.
given: - 5(x + 4)=70.
prove: x = - 18.
statements reasons

• 5(x + 4)=70

-5x - 20 = 70

• 5x - 20 + 20 = 70 + 20
• 5x = 90

\\(\frac{-5x}{-5}=\frac{90}{-5}\\)
x = - 18

1. given: point s is between points r and t and rs = 2x - 11, st = 9, and rt = x + 5

prove: x = 7
statements reasons

1. 1.
2. rs + st = rt 2.
3. rs = 2x - 11,st = 9, and rt = x + 5 3.
4. 4. substitution
5. 5. substitution
6. x - 2 = 5 6.
7. 7.

geometry unit 1.1

Explanation:

Step1: Expand the left - hand side

Using the distributive property \(a(b + c)=ab+ac\), for \(-5(x + 4)\), we get \(-5x-20 = 70\).

Step2: Add 20 to both sides

To isolate the term with \(x\), we use the addition property of equality. \(-5x-20 + 20=70 + 20\), which simplifies to \(-5x=90\).

Step3: Divide both sides by - 5

Using the division property of equality, \(\frac{-5x}{-5}=\frac{90}{-5}\), so \(x=-18\).

For the second problem:

Step1: State the given fact

Point \(S\) is between points \(R\) and \(T\). Reason: Given.

Step2: Use the segment addition postulate

If \(S\) is between \(R\) and \(T\), then \(RS + ST=RT\). Reason: Segment addition postulate.

Step3: State the given lengths

\(RS = 2x-11\), \(ST = 9\), and \(RT=x + 5\). Reason: Given.

Step4: Substitute the lengths into the equation

\((2x-11)+9=x + 5\). Reason: Substitution.

Step5: Simplify the left - hand side

Combine like terms: \(2x-11 + 9=2x-2\), so \(2x-2=x + 5\). Reason: Simplification.

Step6: Subtract \(x\) from both sides

\(2x-x-2=x-x + 5\), which gives \(x-2=5\). Reason: Subtraction property of equality.

Step7: Add 2 to both sides

\(x-2+2=5+2\), so \(x = 7\). Reason: Addition property of equality.

Answer:

1.

Statements	Reasons
\(-5x-20 = 70\)	Distributive property
\(-5x-20 + 20=70 + 20\)	Addition property of equality
\(-5x=90\)	Simplification
\(\frac{-5x}{-5}=\frac{90}{-5}\)	Division property of equality
\(x=-18\)	Simplification

2.

Statements	Reasons
\(RS + ST=RT\)	Segment addition postulate
\(RS = 2x-11\), \(ST = 9\), \(RT=x + 5\)	Given
\((2x-11)+9=x + 5\)	Substitution
\(2x-2=x + 5\)	Simplification
\(x-2=5\)	Subtraction property of equality
\(x = 7\)	Addition property of equality