Question

consider the following system of equations
x+3y=9
2x+6y=30
1) is (0,3) a solution to the system of equations? explain
2) solve the system of equations using elimination

Answer 1

To determine whether the point (0, 3) is a solution to the system of equations, we need to substitute \( x = 0 \) and \( y = 3 \) into both equations and see if both equations are satisfied.

**1) Checking if (0, 3) is a solution:**

The system of equations is:
1. \( x + 3y = 9 \)
2. \( 2x + 6y = 30 \)

Let's substitute \( x = 0 \) and \( y = 3 \) into both equations.

**For the first equation:**
\[
0 + 3(3) = 9 \\
0 + 9 = 9 \\
9 = 9 \quad \text{(True)}
\]

**For the second equation:**
\[
2(0) + 6(3) = 30 \\
0 + 18 = 30 \\
18 = 30 \quad \text{(False)}
\]

Since the second equation does not hold true when we substitute (0, 3), we conclude that (0, 3) is **not** a solution to the system of equations.

---

**2) Solving the system of equations using elimination:**

The system is:
1. \( x + 3y = 9 \) (Equation 1)
2. \( 2x + 6y = 30 \) (Equation 2)

To use elimination, we can observe that Equation 2 is a multiple of Equation 1. In fact, if we multiply Equation 1 by 2, we get:
\[
2(x + 3y) = 2(9) \\
2x + 6y = 18 \quad \text{(This is Equation 1 multiplied by 2)}
\]

Now we have:
1. \( 2x + 6y = 18 \) (modified Equation 1)
2. \( 2x + 6y = 30 \) (Equation 2)

Now we can subtract the modified Equation 1 from Equation 2:
\[
(2x + 6y) - (2x + 6y) = 30 - 18 \\
0 = 12
\]

This is a contradiction (0 cannot equal 12), which indicates that there are no solutions to the system of equations. The two equations represent parallel lines that never intersect.

Thus, the final result is that the system has **no solution**.