Question

Justify your steps using the properties of equality and determine the number of solutions to the equation 3/4(4-8x)+5x=1-7x

Answer 1

To solve the equation \( \frac{3}{4}(4 - 8x) + 5x = 1 - 7x \), we will justify our steps using properties of equality and determine the number of solutions.

### Step 1: Distribute the \( \frac{3}{4} \)
Using the distributive property of equality, we first distribute \( \frac{3}{4} \) across the expression \( (4 - 8x) \).

\[
\frac{3}{4} \cdot 4 - \frac{3}{4} \cdot 8x + 5x = 1 - 7x
\]

Calculating \( \frac{3}{4} \cdot 4 \) gives us 3, and calculating \( \frac{3}{4} \cdot 8x \) gives us \( 6x \).

\[
3 - 6x + 5x = 1 - 7x
\]

### Step 2: Combine like terms on the left side
Now combine the like terms \( -6x + 5x \):

\[
3 - x = 1 - 7x
\]

### Step 3: Isolate the variable \( x \)
Next, we want to isolate \( x \). To do this, we can add \( 7x \) to both sides (using the addition property of equality):

\[
3 - x + 7x = 1 - 7x + 7x
\]

This simplifies to:

\[
3 + 6x = 1
\]

### Step 4: Isolate the constant term
Next, subtract 3 from both sides (again applying the addition property of equality):

\[
6x = 1 - 3
\]

This simplifies to:

\[
6x = -2
\]

### Step 5: Solve for \( x \)
Now, divide both sides by 6 (using the division property of equality):

\[
x = \frac{-2}{6}
\]

This simplifies to:

\[
x = \frac{-1}{3}
\]

### Conclusion: Determine the number of solutions
The equation \( 3/4(4 - 8x) + 5x = 1 - 7x \) has been solved and we found one solution:

\[
x = -\frac{1}{3}
\]

Since we arrived at one specific value for \( x \) through valid algebraic manipulation, we conclude that there is exactly **one solution** to the equation.