Solve the equation using the substitution method: 2m + n = 0 and m + 2n = 3.
The Correct Answer and Explanation is:
Correct Answer:
We are asked to solve the system using the substitution method:
Equation 1:
2m+n=02m + n = 02m+n=0
Equation 2:
m+2n=3m + 2n = 3m+2n=3
Step 1: Solve one equation for one variable
From Equation 1:
2m+n=02m + n = 02m+n=0
Solve for nnn:
n=−2mn = -2mn=−2m
Step 2: Substitute into the second equation
Substitute n=−2mn = -2mn=−2m into Equation 2:
m+2n=3m + 2n = 3m+2n=3
m+2(−2m)=3m + 2(-2m) = 3m+2(−2m)=3
m−4m=3m – 4m = 3m−4m=3
−3m=3-3m = 3−3m=3
Divide both sides by −3-3−3:
m=−1m = -1m=−1
Step 3: Find nnn
We already found that n=−2mn = -2mn=−2m, so:
n=−2(−1)=2n = -2(-1) = 2n=−2(−1)=2
Final Solution:
m=−1m = -1m=−1, n=2n = 2n=2
Explanation:
The substitution method is a common algebraic technique for solving systems of equations. It involves isolating one variable in one equation and substituting its value into the other equation. This converts the system of two equations with two variables into a single equation with one variable, making it easier to solve.
In this problem, the first equation 2m+n=02m + n = 02m+n=0 was straightforward to rearrange for nnn, giving n=−2mn = -2mn=−2m. This expression was then substituted into the second equation, effectively reducing the system to an equation with only mmm.
After simplifying the second equation, we obtained −3m=3-3m = 3−3m=3, which was solved by dividing both sides by −3-3−3, resulting in m=−1m = -1m=−1. Once mmm was known, finding nnn was simple using the earlier expression n=−2mn = -2mn=−2m.
The advantage of the substitution method is its simplicity when one of the equations is easy to rearrange. It provides exact values for the variables, showing where the two lines represented by the equations intersect.
Thus, the solution to the system is:
m=−1m = -1m=−1 and n=2n = 2n=2.
