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Problem 192

# Solve the equation $$(3 / 4) \mathrm{x}+(7 / 8)+1=0$$

Expert verified
The solution to the equation $$(\frac{3}{4})x + (\frac{7}{8}) + 1 = 0$$ is $$x = -\frac{5}{2}$$.
See the step by step solution

## Step 1: Combine the constants

First, let's combine the constants $$\frac{7}{8}$$ and $$1$$. To do this, we need a common denominator. Since the denominator of the fraction $$\frac{7}{8}$$ is already $$8$$, we will convert $$1$$ to a fraction with the same denominator: $$1 = \frac{8}{8}$$. Now we can add the fractions: $$\frac{7}{8} + \frac{8}{8} = \frac{15}{8}$$. The equation becomes: $$(\frac{3}{4})x + \frac{15}{8} = 0$$.

## Step 2: Isolate the x term

Next, we need to isolate the x term, $$(\frac{3}{4})x$$, by moving $$\frac{15}{8}$$ to the other side of the equation. We can do this by subtracting $$\frac{15}{8}$$ from both sides: $(\frac{3}{4})x + \frac{15}{8} - \frac{15}{8} = 0 - \frac{15}{8}$ This simplifies to: $(\frac{3}{4})x = -\frac{15}{8}$

## Step 3: Solve for x

Now all that's left is to solve for x by dividing both sides of the equation by $$\frac{3}{4}$$. To do this, we can instead multiply by the reciprocal of $$\frac{3}{4}$$, which is $$\frac{4}{3}$$: $(\frac{3}{4})x \cdot \frac{4}{3} = -\frac{15}{8} \cdot \frac{4}{3}$ This simplifies to: $x = -\frac{15}{8} \cdot \frac{4}{3}$

## Step 4: Simplify the equation

Now, let's simplify the equation to find the solution for x. To do this, we will multiply the fractions: $-\frac{15}{8} \cdot \frac{4}{3} = -\frac{15 \times 4}{8 \times 3}$ This simplifies to: $x = -\frac{60}{24}$

## Step 5: Reduce the fraction

Finally, we need to reduce the fraction $$-\frac{60}{24}$$ to its simplest form. The greatest common divisor (GCD) of $$60$$ and $$24$$ is $$12$$, so we can divide both the numerator and the denominator by $$12$$: $x = -\frac{60 \div 12}{24 \div 12}$ This simplifies to: $x = -\frac{5}{2}$ So, the solution to the equation $$(\frac{3}{4})x + (\frac{7}{8}) + 1 = 0$$ is $$x = -\frac{5}{2}$$.

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