QuestionSolve Equation $\frac{1}{2} x+\frac{3}{4} x=5+2.5 x$

Studdy Solution

STEP 1

1. The equation involves fractions and decimals.
2. We are solving for the variable $x$ .
3. The equation can be simplified and solved using basic algebraic operations.

STEP 2

1. Combine like terms on the left side of the equation.
2. Move all terms involving $x$ to one side of the equation.
3. Solve for $x$ by isolating it.
4. Check the solution by substituting it back into the original equation.

STEP 3

Combine the like terms on the left side of the equation:
$\frac{1}{2}x + \frac{3}{4}x = \frac{2}{4}x + \frac{3}{4}x = \frac{5}{4}x$
The equation now becomes:
$\frac{5}{4}x = 5 + 2.5x$

STEP 4

Move all terms involving $x$ to one side of the equation. Subtract $2.5x$ from both sides:
$\frac{5}{4}x - 2.5x = 5$
Convert $2.5$ to a fraction to make subtraction easier: $2.5 = \frac{5}{2}$ .
$\frac{5}{4}x - \frac{5}{2}x = 5$

STEP 5

Find a common denominator to combine the terms:
The common denominator of 4 and 2 is 4.
$\frac{5}{4}x - \frac{10}{4}x = 5$
Combine the terms:
$-\frac{5}{4}x = 5$

STEP 6

Solve for $x$ by isolating it. Divide both sides by $-\frac{5}{4}$ :
$x = \frac{5}{-\frac{5}{4}}$
Simplify the division:
$x = 5 \times -\frac{4}{5}$
$x = -4$

STEP 7

Check the solution by substituting $x = -4$ back into the original equation:
$\frac{1}{2}(-4) + \frac{3}{4}(-4) = 5 + 2.5(-4)$
Calculate each term:
Left side:
$\frac{1}{2}(-4) = -2$
$\frac{3}{4}(-4) = -3$
$-2 - 3 = -5$
Right side:
$5 + 2.5(-4) = 5 - 10 = -5$
Both sides are equal, confirming the solution is correct.
The solution is:
$\boxed{-4}$

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