Question

for each of the following systems of equations, indicate whether the point (-1, 4) is a solution yes no answers chosen no no answers chosen possible answers y = x + 5 and y = 5 + x y = -x + 3 and y = -7x - 3 y = 8x + 4 and y = 3 - x

Explanation:

To determine if \((-1, 4)\) is a solution for each system, we substitute \(x = -1\) and \(y = 4\) into both equations of the system.

System 1: \(y = x + 5\) and \(y = 5 + x\)

Step 1: Substitute into \(y = x + 5\)

Substitute \(x = -1\) and \(y = 4\) into \(y = x + 5\):
\(4 = -1 + 5\)
\(4 = 4\) (True)

Step 2: Substitute into \(y = 5 + x\)

Substitute \(x = -1\) and \(y = 4\) into \(y = 5 + x\):
\(4 = 5 + (-1)\)
\(4 = 4\) (True)
Since both equations are satisfied, \((-1, 4)\) is a solution for this system. So we put this system under "Yes".

System 2: \(y = -x + 3\) and \(y = -7x - 3\)

Step 1: Substitute into \(y = -x + 3\)

Substitute \(x = -1\) and \(y = 4\) into \(y = -x + 3\):
\(4 = -(-1) + 3\)
\(4 = 1 + 3\)
\(4 = 4\) (True)

Step 2: Substitute into \(y = -7x - 3\)

Substitute \(x = -1\) and \(y = 4\) into \(y = -7x - 3\):
\(4 = -7(-1) - 3\)
\(4 = 7 - 3\)
\(4 = 4\) (True)
Since both equations are satisfied, \((-1, 4)\) is a solution for this system. So we put this system under "Yes".

System 3: \(y = 8x + 4\) and \(y = 3 - x\)

Step 1: Substitute into \(y = 8x + 4\)

Substitute \(x = -1\) and \(y = 4\) into \(y = 8x + 4\):
\(4 = 8(-1) + 4\)
\(4 = -8 + 4\)
\(4 = -4\) (False)
Since the first equation is not satisfied, we don't need to check the second. \((-1, 4)\) is not a solution for this system. So we put this system under "No".

Final Categorization:

• Yes:
• \(y = x + 5\) and \(y = 5 + x\)
• \(y = -x + 3\) and \(y = -7x - 3\)
• No:
• \(y = 8x + 4\) and \(y = 3 - x\)

Answer:

To determine if \((-1, 4)\) is a solution for each system, we substitute \(x = -1\) and \(y = 4\) into both equations of the system.

System 1: \(y = x + 5\) and \(y = 5 + x\)

Step 1: Substitute into \(y = x + 5\)

Substitute \(x = -1\) and \(y = 4\) into \(y = x + 5\):
\(4 = -1 + 5\)
\(4 = 4\) (True)

Step 2: Substitute into \(y = 5 + x\)

Substitute \(x = -1\) and \(y = 4\) into \(y = 5 + x\):
\(4 = 5 + (-1)\)
\(4 = 4\) (True)
Since both equations are satisfied, \((-1, 4)\) is a solution for this system. So we put this system under "Yes".

System 2: \(y = -x + 3\) and \(y = -7x - 3\)

Step 1: Substitute into \(y = -x + 3\)

Substitute \(x = -1\) and \(y = 4\) into \(y = -x + 3\):
\(4 = -(-1) + 3\)
\(4 = 1 + 3\)
\(4 = 4\) (True)

Step 2: Substitute into \(y = -7x - 3\)

Substitute \(x = -1\) and \(y = 4\) into \(y = -7x - 3\):
\(4 = -7(-1) - 3\)
\(4 = 7 - 3\)
\(4 = 4\) (True)
Since both equations are satisfied, \((-1, 4)\) is a solution for this system. So we put this system under "Yes".

System 3: \(y = 8x + 4\) and \(y = 3 - x\)

Step 1: Substitute into \(y = 8x + 4\)

Substitute \(x = -1\) and \(y = 4\) into \(y = 8x + 4\):
\(4 = 8(-1) + 4\)
\(4 = -8 + 4\)
\(4 = -4\) (False)
Since the first equation is not satisfied, we don't need to check the second. \((-1, 4)\) is not a solution for this system. So we put this system under "No".

Final Categorization:

• Yes:
• \(y = x + 5\) and \(y = 5 + x\)
• \(y = -x + 3\) and \(y = -7x - 3\)
• No:
• \(y = 8x + 4\) and \(y = 3 - x\)