Sure! Let's solve the equation step-by-step.
The equation we need to solve is:
[tex]\[ -\frac{5}{4} x + 8 = \frac{39}{4} + \frac{1}{2} x \][/tex]
1. Combine like terms:
To eliminate the fractions, we can express both sides with a common denominator. The common denominator for 4 and 2 is 4. Let's rewrite [tex]\(\frac{1}{2} x\)[/tex] as [tex]\(\frac{2}{4} x\)[/tex]:
[tex]\[ -\frac{5}{4} x + 8 = \frac{39}{4} + \frac{2}{4} x \][/tex]
2. Eliminate fractions by multiplying through by 4:
Multiplying every term by 4 to clear the fractions:
[tex]\[ 4 \left(-\frac{5}{4} x + 8\right) = 4 \left( \frac{39}{4} + \frac{2}{4} x \right) \][/tex]
This simplifies to:
[tex]\[ -5x + 32 = 39 + 2x \][/tex]
3. Move all terms involving [tex]\(x\)[/tex] to one side and constant terms to the other side:
Subtract [tex]\(2x\)[/tex] from both sides to get all [tex]\(x\)[/tex]-terms on one side:
[tex]\[ -5x - 2x + 32 = 39 \][/tex]
This simplifies to:
[tex]\[ -7x + 32 = 39 \][/tex]
4. Isolate the variable [tex]\(x\)[/tex]:
Subtract 32 from both sides to get all constant terms on one side:
[tex]\[ -7x + 32 - 32 = 39 - 32 \][/tex]
[tex]\[ -7x = 7 \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Divide both sides by [tex]\(-7\)[/tex]:
[tex]\[ x = \frac{7}{-7} \][/tex]
[tex]\[ x = -1 \][/tex]
So, the solution to the equation is:
[tex]\[ x = -1 \][/tex]
Thus, after following these steps, we find that the value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\(-1\)[/tex].
