Certainly! Let's solve the equation step by step.
We are given the equation:
[tex]\[ x + \frac{2}{7} = 1 \frac{2}{3} + \frac{1}{5} \][/tex]
### Step 1: Convert the Mixed Number to an Improper Fraction
First, let's convert the mixed number [tex]\(1 \frac{2}{3}\)[/tex] to an improper fraction.
[tex]\[ 1 \frac{2}{3} = 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \][/tex]
So, the equation now becomes:
[tex]\[ x + \frac{2}{7} = \frac{5}{3} + \frac{1}{5} \][/tex]
### Step 2: Add the Fractions on the Right-Hand Side
We need to add the fractions [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{1}{5}\)[/tex]. To do this, we first find a common denominator.
The lowest common denominator (LCD) of 3 and 5 is 15. Therefore, we convert the fractions:
[tex]\[
\frac{5}{3} = \frac{5 \times 5}{3 \times 5} = \frac{25}{15}
\][/tex]
[tex]\[
\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}
\][/tex]
Now, we add the fractions:
[tex]\[
\frac{25}{15} + \frac{3}{15} = \frac{25 + 3}{15} = \frac{28}{15}
\][/tex]
So, the equation now becomes:
[tex]\[ x + \frac{2}{7} = \frac{28}{15} \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
To isolate [tex]\( x \)[/tex], we need to subtract [tex]\(\frac{2}{7}\)[/tex] from both sides of the equation.
First, let's find a common denominator for the fractions [tex]\(\frac{28}{15}\)[/tex] and [tex]\(\frac{2}{7}\)[/tex].
The lowest common denominator (LCD) of 15 and 7 is 105. We convert the fractions as follows:
[tex]\[
\frac{28}{15} = \frac{28 \times 7}{15 \times 7} = \frac{196}{105}
\][/tex]
[tex]\[
\frac{2}{7} = \frac{2 \times 15}{7 \times 15} = \frac{30}{105}
\][/tex]
Now, we subtract the fractions:
[tex]\[
x = \frac{196}{105} - \frac{30}{105} = \frac{196 - 30}{105} = \frac{166}{105}
\][/tex]
### Step 4: Simplify the Fraction (if possible)
The fraction [tex]\(\frac{166}{105}\)[/tex] can be simplified if there is a common factor of 166 and 105.
However, 166 and 105 do not have a common factor other than 1, so the fraction cannot be simplified further.
### Final Step
To convert [tex]\(\frac{166}{105}\)[/tex] into a decimal form for better understanding, we perform the division:
[tex]\[ \frac{166}{105} \approx 1.580952380952381 \][/tex]
Thus, the solution to the equation is:
[tex]\[ x \approx 1.580952380952381 \][/tex]
