Question

regents questions

1. solve for x: 15x - 3(3x + 4) = 6
2. 1
3. $\frac{1}{2}$
4. 3
5. $\frac{1}{3}$
6. if -2x + 3 = 7 and 3x + 1 = 5 + y, the value of y is
7. 1
8. 0
9. -10
10. 10

challenge questions

1. (x + 6) - (2x + 7) - 3x = -9
2. (4x + 2) - (12x + 8) + 2(5x - 3) = 16

Explanation:

Problem 18: Solve for \( x \): \( 15x - 3(3x + 4) = 6 \)

Step 1: Distribute the -3

First, we distribute the \(-3\) into the parentheses \( 3x + 4 \). Using the distributive property \( a(b + c) = ab + ac \), we get:
\( 15x - 9x - 12 = 6 \)

Step 2: Combine like terms

Next, we combine the \( x \)-terms \( 15x - 9x \). This simplifies to:
\( 6x - 12 = 6 \)

Step 3: Add 12 to both sides

To isolate the term with \( x \), we add 12 to both sides of the equation:
\( 6x - 12 + 12 = 6 + 12 \)
Simplifying both sides gives:
\( 6x = 18 \)

Step 4: Divide by 6

Finally, we divide both sides by 6 to solve for \( x \):
\( \frac{6x}{6} = \frac{18}{6} \)
Which simplifies to:
\( x = 3 \)

Step 1: Solve for \( x \) from \( -2x + 3 = 7 \)

First, we solve the equation \( -2x + 3 = 7 \) for \( x \). Subtract 3 from both sides:
\( -2x + 3 - 3 = 7 - 3 \)
Simplifying gives:
\( -2x = 4 \)
Divide both sides by \(-2\):
\( \frac{-2x}{-2} = \frac{4}{-2} \)
So, \( x = -2 \)

Step 2: Substitute \( x = -2 \) into \( 3x + 1 = 5 + y \)

Now, substitute \( x = -2 \) into the second equation \( 3x + 1 = 5 + y \):
\( 3(-2) + 1 = 5 + y \)
Calculate the left side: \( -6 + 1 = -5 \), so:
\( -5 = 5 + y \)

Step 3: Solve for \( y \)

Subtract 5 from both sides:
\( -5 - 5 = 5 + y - 5 \)
Simplifying gives:
\( -10 = y \)

Step 1: Remove parentheses

First, we remove the parentheses. Remember that the negative sign in front of \( (2x + 7) \) will distribute:
\( x + 6 - 2x - 7 - 3x = -9 \)

Step 2: Combine like terms

Next, we combine the \( x \)-terms and the constant terms:
For \( x \)-terms: \( x - 2x - 3x = -4x \)
For constant terms: \( 6 - 7 = -1 \)
So the equation becomes:
\( -4x - 1 = -9 \)

Step 3: Add 1 to both sides

Add 1 to both sides to isolate the term with \( x \):
\( -4x - 1 + 1 = -9 + 1 \)
Simplifying gives:
\( -4x = -8 \)

Step 4: Divide by -4

Divide both sides by \(-4\) to solve for \( x \):
\( \frac{-4x}{-4} = \frac{-8}{-4} \)
Which simplifies to:
\( x = 2 \)

Answer:

\( x = 3 \) (corresponding to option 3)

Problem 19: If \( -2x + 3 = 7 \) and \( 3x + 1 = 5 + y \), the value of \( y \) is