Question

Solve the system by elimination 8N +6 minus 3M equals zero. 32 equals M -N

Answer 1

To solve the system of equations using elimination, let's first rewrite the given equations clearly:

1. \( 8N + 6 - 3M = 0 \)
2. \( 32 = M - N \)

Now, let's rearrange each equation into a more standard form.

For equation 1:
\[
8N - 3M + 6 = 0 \quad \text{(we can subtract 6 from both sides)}
\]
\[
8N - 3M = -6 \quad \text{(Equation 1)}
\]

For equation 2:
\[
M - N = 32 \quad \text{(we can rearrange this as)}
\]
\[
M = N + 32 \quad \text{(Equation 2)}
\]

Now, we can substitute Equation 2 into Equation 1.

Substituting \( M = N + 32 \) into \( 8N - 3M = -6 \):

\[
8N - 3(N + 32) = -6
\]
Expanding this gives:
\[
8N - 3N - 96 = -6
\]

Combining like terms:
\[
5N - 96 = -6
\]
Adding 96 to both sides:
\[
5N = 90
\]
Dividing both sides by 5:
\[
N = 18
\]

Now that we have \( N \), we can find \( M \) using Equation 2:
\[
M = N + 32 = 18 + 32 = 50
\]

Thus, the solution to the system of equations is:
\[
N = 18, \quad M = 50
\]

In summary, the solution is:
\[
\boxed{(18, 50)}
\]

Answer 2

Did you solve it using elimination?

Answer 3

I apologize for the confusion; I used substitution in my previous solution instead of elimination. Let’s resolve the system of equations by elimination.

Given the equations:
1. \( 8N + 6 - 3M = 0 \)
2. \( M - N = 32 \)

Let's start by rewriting them in standard form:

1. Rearranging the first equation:
\[
8N - 3M + 6 = 0 \quad \Rightarrow \quad 8N - 3M = -6
\]

2. Rearranging the second equation:
\[
M - N = 32 \quad \Rightarrow \quad -N + M = 32 \quad \Rightarrow \quad -1N + 1M = 32
\]

Now we have:

1. \( 8N - 3M = -6 \) (Equation 1)
2. \( -N + M = 32 \) (Equation 2)

To eliminate one of the variables, we can manipulate these equations. We will eliminate \( M \). For this, let’s multiply Equation 2 by 3 so that the coefficients of \( M \) will match:

\[
3(-N + M) = 3(32) \quad \Rightarrow \quad -3N + 3M = 96 \quad \text{(Equation 3)}
\]

Now, our system looks like this:

1. \( 8N - 3M = -6 \) (Equation 1)
2. \( -3N + 3M = 96 \) (Equation 3)

Next, we add Equations 1 and 3 together:

\[
(8N - 3M) + (-3N + 3M) = -6 + 96
\]

This simplifies to:

\[
5N = 90
\]

Dividing both sides by 5 gives:

\[
N = 18
\]

Now that we have \( N \), we can substitute it back into Equation 2 to find \( M \):

Using Equation 2:
\[
M - 18 = 32
\]

Solving for \( M \):
\[
M = 32 + 18 = 50
\]

Thus, the solution to the system of equations is:

\[
N = 18, \quad M = 50
\]

Final answer:
\[
\boxed{(18, 50)}
\]