Question

linear systems
identifying solutions to a system of linear equations
for each ordered pair, determine whether it is a solution to the system of equations.
\

$$\begin{cases} -14x + 2y = -6 \\\\ y = 7x - 3 \\end{cases}$$

(x, y)	is it a solution?
(0, -3)	∘ (yes)	∘ (no)
(2, 11)	∘ (yes)	∘ (no)
(-1, 7)	∘ (yes)	∘ (no)

Explanation:

Step1: Recall solution definition

A solution to a system of equations satisfies all equations in the system. We'll substitute each ordered pair \((x,y)\) into both equations and check if they hold.

Step2: Check \((2, -4)\)

Equation 1: \(-14x + 2y = -6\)

Substitute \(x = 2\), \(y = -4\):
\(-14(2) + 2(-4) = -28 - 8 = -36\).
\(-36
eq -6\), so it fails Equation 1.

Equation 2: \(y = 7x - 3\)

Substitute \(x = 2\): \(7(2) - 3 = 14 - 3 = 11\).
\(11
eq -4\), so it fails Equation 2.
Conclusion: No.

Step3: Check \((0, -3)\)

Equation 1: \(-14(0) + 2(-3) = 0 - 6 = -6\).

This matches \(-6\).

Equation 2: \(7(0) - 3 = -3\).

This matches \(y = -3\).
Conclusion: Yes.

Step4: Check \((2, 11)\)

Equation 1: \(-14(2) + 2(11) = -28 + 22 = -6\).

This matches \(-6\).

Equation 2: \(7(2) - 3 = 14 - 3 = 11\).

This matches \(y = 11\).
Conclusion: Yes.

Step5: Check \((-1, 7)\)

Equation 1: \(-14(-1) + 2(7) = 14 + 14 = 28\).

\(28
eq -6\), so it fails Equation 1.

Equation 2: \(7(-1) - 3 = -7 - 3 = -10\).

\(-10
eq 7\), so it fails Equation 2.
Conclusion: No.

Answer:

• \((2, -4)\): No
• \((0, -3)\): Yes
• \((2, 11)\): Yes
• \((-1, 7)\): No