Sure, let's solve the given equation step-by-step:
The given equation is:
[tex]\[
\frac{3}{4} x + 3 - 2 x = -\frac{1}{4} + \frac{1}{2} x + 5
\][/tex]
### Step 1: Combine like terms on both sides of the equation.
First, let's combine the terms involving [tex]\( x \)[/tex] on the left side:
[tex]\[
\frac{3}{4} x - 2 x + 3
\][/tex]
This can be rewritten as:
[tex]\[
\left( \frac{3}{4} - 2 \right)x + 3
\][/tex]
Next, combine the terms involving [tex]\( x \)[/tex] on the right side:
[tex]\[
-\frac{1}{4} + \frac{1}{2} x + 5
\][/tex]
### Step 2: Simplify the coefficients of [tex]\( x \)[/tex].
[tex]\[
\frac{3}{4} - 2 = \frac{3}{4} - \frac{8}{4} = \frac{3 - 8}{4} = -\frac{5}{4}
\][/tex]
So, the equation now looks like this:
[tex]\[
-\frac{5}{4} x + 3 = -\frac{1}{4} + \frac{1}{2} x + 5
\][/tex]
### Step 3: Move all terms involving [tex]\( x \)[/tex] to one side and the constant terms to the other side.
Subtract [tex]\(\frac{1}{2} x\)[/tex] from both sides:
[tex]\[
-\frac{5}{4} x - \frac{1}{2} x + 3 = -\frac{1}{4} + 5
\][/tex]
Simplify the left side:
[tex]\[
-\frac{5}{4} x - \frac{2}{4} x = -\frac{7}{4} x
\][/tex]
And combine the constants on the right side:
[tex]\[
-\frac{1}{4} + 5 = 4.75
\][/tex]
So, the equation now is:
[tex]\[
-\frac{7}{4} x + 3 = 4.75
\][/tex]
### Step 4: Simplify further by subtracting 3 from both sides.
[tex]\[
-\frac{7}{4} x = 4.75 - 3
\][/tex]
[tex]\[
-\frac{7}{4} x = 1.75
\][/tex]
### Step 5: Solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex].
Multiply both sides by [tex]\(-4/7\)[/tex]:
[tex]\[
x = 1.75 \times \left(\frac{-4}{7}\right)
\][/tex]
### Final result:
[tex]\[
x = -1.4
\][/tex]
So the solution to the equation [tex]\(\frac{3}{4} x + 3 - 2 x = -\frac{1}{4} + \frac{1}{2} x + 5\)[/tex] is:
[tex]\[
x = -1.4
\][/tex]
The process ensures we carefully combined like terms, applied arithmetic properties, and isolated the variable to find the solution.
