To eliminate the fractions in the given equation:
[tex]\[
-\frac{3}{4}m - \frac{1}{2} = 2 + \frac{1}{4}m
\][/tex]
we need to find an appropriate number to multiply each term by so that the equation is free of fractions.
1. Identify the denominators.
The denominators in the equation are 4, 2, and 4.
2. Find the least common multiple (LCM) of these denominators.
- The denominators are 4, 2, and 4 again.
- The LCM of 2 and 4 is 4 since 4 is the smallest number that both 2 and 4 divide into evenly.
3. Multiply each term in the equation by the LCM (which is 4):
Multiplying by 4 will eliminate all the fractions:
[tex]\[
4 \left(-\frac{3}{4}m\right) - 4 \left(\frac{1}{2}\right) = 4 \cdot 2 + 4 \left(\frac{1}{4}m\right)
\][/tex]
4. Simplify each term:
- [tex]\( 4 \left(-\frac{3}{4}m\right) = -3m \)[/tex]
- [tex]\( 4 \left(\frac{1}{2}\right) = 2 \)[/tex]
- [tex]\( 4 \cdot 2 = 8 \)[/tex]
- [tex]\( 4 \left(\frac{1}{4}m\right) = m \)[/tex]
5. Rewrite the equation without fractions:
[tex]\[
-3m - 2 = 8 + m
\][/tex]
By multiplying each term of the original equation by 4, we have successfully eliminated the fractions.
Therefore, the number that can be used to multiply each term of the equation to eliminate the fractions is:
[tex]\[
\boxed{4}
\][/tex]
