Question

1. given: 11x - 6y = -1; x = 8

prove: \\(\frac{89}{6} = y\\)

11x - 6y = -1; x = 8 \quad a. \\_\\_\\_\\_\\_\\_
88 - 6y = -1 \quad \quad \quad b. \\_\\_\\_\\_\\_\\_
-6y = -89 \quad \quad \quad \quad c. \\_\\_\\_\\_\\_\\_
y = \\(\frac{89}{6}\\) \quad \quad \quad \quad \quad d. \\_\\_\\_\\_\\_\\_
\\(\frac{89}{6} = y\\) \quad \quad \quad \quad \quad e. \\_\\_\\_\\_\\_\\_

a.
a. given
b. symmetric property of equality
c. subtraction property of equality
d. division property of equality
e. reflexive property of equality

b.
a. given
b. substitution property
c. subtraction property of equality
d. division property of equality
e. symmetric property of equality

c.
a. given
b. substitution property
c. subtraction property of equality
d. division property of equality
e. reflexive property of equality

d.
a. given
b. substitution property
c. subtraction property of equality
d. addition property of equality
e. symmetric property of equality

Explanation:

Step1: Analyze Step a

The first step is just stating the given equations, so it's the Given property. So for part a, the answer is a. Given.

Step2: Analyze Step b

We substitute \(x = 8\) into \(11x-6y=-1\) to get \(88 - 6y=-1\). This is the Substitution Property (replacing \(x\) with its value). So for part b, the answer is b. Substitution Property.

Step3: Analyze Step c

From \(88-6y = - 1\) to \(-6y=-89\), we subtract 88 from both sides. This is the Subtraction Property of Equality (subtracting the same number from both sides of an equation). So for part c, the answer is c. Subtraction Property of Equality.

Step4: Analyze Step d

From \(-6y=-89\) to \(y=\frac{89}{6}\), we divide both sides by \(- 6\) (or multiply by \(-\frac{1}{6}\)). This is the Division Property of Equality (dividing both sides of an equation by the same non - zero number). So for part d, the answer is d. Division Property of Equality.

Step5: Analyze Step e

From \(y = \frac{89}{6}\) to \(\frac{89}{6}=y\), we are using the Symmetric Property of Equality (if \(a = b\), then \(b=a\)). So for part e, the answer is e. Symmetric Property of Equality.

Now, looking at the options:

• For part a: Option a (a. Given) is correct.
• For part b: Option b (b. Substitution Property) is correct.
• For part c: Option c (c. Subtraction Property of Equality) is correct.
• For part d: Option d (d. Division Property of Equality) is correct.
• For part e: Option e (e. Symmetric Property of Equality) is correct. But looking at the option sets:
• The set for part b (option b) has: a. Given, b. Substitution Property, c. Subtraction Property of Equality, d. Division Property of Equality, e. Symmetric Property of Equality. Which matches our analysis for each step.

Answer:

b. a. Given; b. Substitution Property; c. Subtraction Property of Equality; d. Division Property of Equality; e. Symmetric Property of Equality