Answer:
m = -0.6 or m = -3/5
Explanation:
Step 1: Convert 1 2/5, 2 5/6, and -3 1/2 (mixed numbers) to improper fractions:
- Converting 1 2/5, 2 5/6 and -3 1/2 to improper fractions will make it easier when solving for m.
- When converting mixed numbers to improper fractions, we keep the same denominator.
- For the numerator, we multiply the whole number by the original denominator and add the original numerator.
- For -3 1/2, we can leave off the negative and add it back at the end.
- Since all three numbers are coefficients of the variable m, we can multiply the improper fractions by m.
Converting 1 2/5 to an improper fraction:
((1 * 5) + 2) / 5
(5 + 2) / 5
7/5m
Converting 2 5/6 to an improper fraction:
((2 * 6) + 5) / 6
(12 + 5) / 6
17/6m
Converting -3 1/2 to an improper fraction:
-(((3 * 2) + 1) / 2)
-((6 + 1) / 2)
-7/2m
Thus, our new equation with the improper fractions is given by:
7/5m + 16/25 = 17/6m - 3/5 - 7/2m
Step 2: Eliminate the fractions by finding the least common denominator (LCD):
- The least common denominator (LCD) is the essentially the least common multiple (LCM) of the denominators.
- Finding it allows us to clear all the fractions.
- The least common denominator is 150.
Now we can eliminate the fractions by multiplying each fraction by 150 and simplifying:
150(7/5m + 16/25 = 17/6m - 3/5 - 7/2m)
1050/5m + 2400/25 = 2550/6m - 450/5 - 1050/2m
210m + 96 = 425m - 90 - 525m
Step 3: Combine like terms on the right-hand side:
210m + 96 = (425m - 525m) - 90
210m + 96 = -100m - 90
Step 4: Add 90 to both sides:
(210m + 96 = -100m - 90) + 90
210m + 186 = -100m
Step 5: Subtract 210m from both sides:
(210m + 186 = -100m) - 210m
186 = -310m
Step 6: Divide both sides by -310 to solve for m:
(186 = -310m) / -310
m = -0.6 or m = -3/5
Thus, m = -0.6 or -3/5.
You can use either the decimal answer or the fractional answer, depending on your personal preference.
Optional Step 7: Check the validity of the answer:
We can check that our answer is correct by plugging in -0.6 for m and seeing if we get the same answer on both sides:
7/5(-0.6) + 16/25 = 17/6(-0.6) - 3/5 - 7/2(-0.6)
-21/25 + 16/25 = -17/10 - 3/5 + 21/10
-5/25 = -23/10 + 21/10
-1/5 = -2/10
-1/5 = -1/5
-0.2 = -0.2
Thus, our answer is correct.