Which value of $x$ satisfies the equation $\frac{4}{5}\left(x+\frac{2}{3}\right)=\frac{32}{15}$ ? $\newline$ $-2$ $\newline$ $2$ $\newline$ $-1$ $\newline$ $1$

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Grade 8

Add and subtract rational numbers

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Q. Which value of

x

satisfies the equation

\frac{4}{5}\left(x+\frac{2}{3}\right)=\frac{32}{15}

\newline

-2

\newline

2

\newline

-1

\newline

1

Write Equation: Write down the equation. $\newline$ $(\frac{4}{5})(x + \frac{2}{3}) = \frac{32}{15}$ $\newline$ We need to find the value of $x$ that satisfies this equation.
Multiply by Reciprocal: Multiply both sides of the equation by the reciprocal of $\frac{4}{5}$ to isolate the term with $x$ . The reciprocal of $\frac{4}{5}$ is $\frac{5}{4}$ . So we multiply both sides by $\frac{5}{4}$ to cancel out the $\frac{4}{5}$ on the left side. $(\frac{5}{4}) \times (\frac{4}{5})(x + \frac{2}{3}) = (\frac{5}{4}) \times (\frac{32}{15})$
Simplify Equation: Simplify both sides of the equation. $\newline$ On the left side, $(\frac{5}{4}) \times (\frac{4}{5}) = 1$ , so we are left with $x + \frac{2}{3}$ . $\newline$ On the right side, we multiply the numerators and denominators: $(5 \times 32) / (4 \times 15)$ . $\newline$ $x + \frac{2}{3} = (\frac{160}{60})$
Simplify Fraction: Simplify the fraction on the right side of the equation. $\newline$ $\frac{160}{60}$ simplifies to $\frac{8}{3}$ because both $160$ and $60$ are divisible by $20$ . $\newline$ $x + \frac{2}{3} = \frac{8}{3}$
Subtract to Solve: Subtract $\frac{2}{3}$ from both sides to solve for $x$ . $\newline$ $x + \frac{2}{3} - \frac{2}{3} = \frac{8}{3} - \frac{2}{3}$ $\newline$ $x = \frac{8}{3} - \frac{2}{3}$
Subtract Fractions: Subtract the fractions on the right side. $\newline$ Since the denominators are the same, we can subtract the numerators directly. $\newline$ $x = \frac{8 - 2}{3}$ $\newline$ $x = \frac{6}{3}$
Simplify to Find Value: Simplify the fraction to find the value of $x$ . $\frac{6}{3}$ simplifies to $2$ because $6$ divided by $3$ equals $2$ . $x = 2$