Question

learning goals from lesson 1.3
i can solve linear equations in one variable that have variables on both sides. (standards: a.ced.1, a.rei.3) i got it!

1. solve $27q - 10 = 72q - 100$.

a. $q = 2$
b. $q = -90$
c. $q = -2$
d. $q = 90$

Explanation:

Step1: Move variables to one side

Subtract \(27q\) from both sides: \(27q - 10 - 27q = 72q - 100 - 27q\), simplifying to \(-10 = 45q - 100\).

Step2: Move constants to the other side

Add 100 to both sides: \(-10 + 100 = 45q - 100 + 100\), simplifying to \(90 = 45q\).

Step3: Solve for \(q\)

Divide both sides by 45: \(\frac{90}{45} = \frac{45q}{45}\), so \(q = 2\)? Wait, no, wait—wait, let's check again. Wait, original equation: \(27q - 10 = 72q - 100\). Subtract \(27q\): \(-10 = 45q - 100\). Add 100: \(90 = 45q\). Then \(q = 90/45 = 2\)? But wait, let's plug back in. Wait, maybe I made a mistake. Wait, \(27(2) -10 = 54 -10 = 44\). \(72(2) -100 = 144 -100 = 44\). Oh, right, so \(q=2\) is a. Wait, but let me check the options again. Wait, the options are a. \(q=2\), b. \(q=-90\), c. \(q=-2\), d. \(q=90\). So when we solve \(27q -10 = 72q -100\), subtract \(27q\): \(-10 = 45q -100\), add 100: \(90 = 45q\), so \(q = 2\). So the correct option is a.

Wait, but let me re-express the steps correctly.

Step1: Isolate variable terms

Subtract \(27q\) from both sides:
\(27q - 10 - 27q = 72q - 100 - 27q\)
Simplifies to: \(-10 = 45q - 100\)

Step2: Isolate constant terms

Add \(100\) to both sides:
\(-10 + 100 = 45q - 100 + 100\)
Simplifies to: \(90 = 45q\)

Step3: Solve for \(q\)

Divide both sides by \(45\):
\(\frac{90}{45} = \frac{45q}{45}\)
Simplifies to: \(q = 2\)

Answer:

a. \(q = 2\)