To solve the expression [tex]\(-\frac{12}{7} + \frac{3}{14}\)[/tex], we need to add these two fractions. Here is a detailed, step-by-step solution:
1. Find a common denominator:
The denominators are 7 and 14. The least common multiple (LCM) of 7 and 14 is 14. So, we need to express both fractions with a common denominator of 14.
2. Rewrite each fraction with the common denominator:
[tex]\[
-\frac{12}{7} \quad \text{needs to be converted to a fraction with denominator 14.}
\][/tex]
Multiply both the numerator and the denominator of [tex]\(-\frac{12}{7}\)[/tex] by 2:
[tex]\[
-\frac{12 \times 2}{7 \times 2} = -\frac{24}{14}
\][/tex]
Now, the fractions are:
[tex]\[
-\frac{24}{14} \quad \text{and} \quad \frac{3}{14}
\][/tex]
3. Add the fractions:
Since the denominators are the same, we can add the numerators directly:
[tex]\[
-\frac{24}{14} + \frac{3}{14} = \frac{-24 + 3}{14} = \frac{-21}{14}
\][/tex]
4. Simplify the fraction:
To simplify [tex]\(\frac{-21}{14}\)[/tex], we need to find the greatest common divisor (GCD) of 21 and 14, which is 7.
Divide both the numerator and the denominator by 7:
[tex]\[
\frac{-21 \div 7}{14 \div 7} = \frac{-3}{2}
\][/tex]
So, the simplified form of [tex]\(-\frac{12}{7} + \frac{3}{14}\)[/tex] is [tex]\(-\frac{3}{2}\)[/tex].
Therefore, the solution is:
[tex]\[
-\frac{3}{2}
\][/tex]
