solve the equation 3/4x+3-2x=-1/4+1/2x+5 i combined terms what next?

1 answer

Let's solve the equation step by step.

The given equation is:

\[ \frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5 \]

First, let's combine like terms on both sides of the equation.

Step 1: Simplify the left side

On the left side, combine \(\frac{3}{4}x\) and \(-2x\).

To do this, we convert \(-2x\) to a fraction with a common denominator. We can write \(-2x\) as \(-\frac{8}{4}x\):

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

So the left side is now:

\[ -\frac{5}{4}x + 3 \]

Step 2: Simplify the right side

Now, simplify the right side. First combine \(-\frac{1}{4}\) and \(5\):

\[ -\frac{1}{4} + 5 = -\frac{1}{4} + \frac{20}{4} = \frac{19}{4} \]

Now, combine this with \(\frac{1}{2}x\). Convert \(\frac{1}{2}x\) to have a common denominator with \(\frac{19}{4}\):

\[ \frac{1}{2}x = \frac{2}{4}x \]

So the right side becomes:

\[ \frac{2}{4}x + \frac{19}{4} = \frac{1}{2}x + \frac{19}{4} \]

Step 3: Write the simplified equation

Now we can rewrite the entire equation:

\[ -\frac{5}{4}x + 3 = \frac{1}{2}x + \frac{19}{4} \]

Step 4: Eliminate the fractions

To eliminate the fractions, multiply the whole equation by 4 (the least common denominator):

\[ 4 \left(-\frac{5}{4}x + 3\right) = 4 \left(\frac{1}{2}x + \frac{19}{4}\right) \]

This simplifies to:

\[ -5x + 12 = 2x + 19 \]

Step 5: Solve for \(x\)

Now, combine like terms:

  1. Add \(5x\) to both sides:

\[ 12 = 7x + 19 \]

  1. Subtract \(19\) from both sides:

\[ 12 - 19 = 7x \]

\[ -7 = 7x \]

  1. Divide both sides by \(7\):

\[ x = -1 \]

So the solution to the equation is:

\[ x = -1 \]