Q3. - Algebra 1

Question

Evaluate the following equation and check the solution is correct or not:

$\frac{2}{3}+w=1\frac{1}{2}$

Step-by-Step Solution

Answer

The solution of the equation $\frac{2}{3}+w=1\frac{1}{2}$ is $w=\frac{5}{6}$.

Step 1. Given information.

The given equation is $\frac{2}{3}+w=1\frac{1}{2}$.

Step 2. Write the concept.

In an equation, the variable represents the number that satisfies the equation.

To solve an equation means to find the value of the variable that makes the equation true.

If an equation is true and the same number is added to each side of the equation, then the resulting equivalent equation is also true.

Step 3. Solve the equation.

The equation $\frac{2}{3}+w=1\frac{1}{2}$ can be solved below:

Since, $1\frac{1}{2}=\frac{3}{2}$. Therefore,

$\frac{2}{3}+w=\frac{3}{2}$

Multiply both sides by $\frac{2}{3}$,

$$\begin{gathered} \left(\frac{2}{3}+w\right)\times \frac{2}{3}=\frac{3}{2}\times \frac{2}{3} \\ \left(\frac{2}{3}+w\right)\times \frac{2}{3}=1 \end{gathered}$$

Step 4. Apply the Distributive Property.

The distributive property is defined as in below.

$\left(b+c\right)a=ba+ca$

Here a, b, and c are real numbers.

$$\begin{gathered} \left(\frac{2}{3}+w\right)\times \frac{2}{3}=1 \\ \frac{2}{3}\times \frac{2}{3}+w\times \frac{2}{3}=1 \\ \frac{4}{9}+\frac{2w}{3}=1 \end{gathered}$$

The above equation can be written as in below.

$$\begin{gathered} \frac{4}{9}+\frac{2w\times 3}{3\times 3}=1 \\ \frac{4}{9}+\frac{6w}{9}=1 \\ 4+6w=9 \\ 6w=9-4 \end{gathered}$$

So,

$w=\frac{5}{6}$

Step 5. Check the solution.

Take the equation $\frac{2}{3}+w=1\frac{1}{2}$.

Substitute $w=\frac{5}{6}$ on the right-hand side of the equation.

$$\begin{gathered} \frac{2}{3}+w=1\frac{1}{2} \\ \frac{2}{3}+\frac{5}{6}=1\frac{1}{2} \\ \frac{2\times 2+5}{6}=1\frac{1}{2} \\ \frac{4+5}{6}=1\frac{1}{2} \end{gathered}$$

This shows that,

$\frac{3}{2}=\frac{3}{2}$

This shows that, $w=\frac{5}{6}$ is the solution of the given equation.