Determine if each ordered pair is a solution to the given system.
5 x minus 2 y equals 13
4 x plus 3 y equals 15
a. left parenthesis 3 comma 1 right parenthesis
b. left parenthesis negative 1 comma negative 9 right parenthesis
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Part 1
a. Is the ordered pair left parenthesis 3 comma 1 right parenthesis a solution to the given​ system?
Yes

To verify whether \((3, 1)\) is a solution to the given system of equations:

The system of equations is:
1. \(5x - 2y = 13\)
2. \(4x + 3y = 15\)

### Step 1: Substitute \((x, y) = (3, 1)\) into both equations.

#### Equation 1: \(5x - 2y = 13\)
Substitute \(x = 3\) and \(y = 1\):
\[
5(3) - 2(1) = 15 - 2 = 13
\]
This is true, so \((3, 1)\) satisfies the first equation.

#### Equation 2: \(4x + 3y = 15\)
Substitute \(x = 3\) and \(y = 1\):
\[
4(3) + 3(1) = 12 + 3 = 15
\]
This is also true, so \((3, 1)\) satisfies the second equation.

### Conclusion for Part (a)
Since \((3, 1)\) satisfies both equations, **the ordered pair \((3, 1)\) is indeed a solution to the system.**

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### Part (b): Determine if \((-1, -9)\) is a solution.

#### Step 1: Substitute \((x, y) = (-1, -9)\) into both equations.

#### Equation 1: \(5x - 2y = 13\)
Substitute \(x = -1\) and \(y = -9\):
\[
5(-1) - 2(-9) = -5 + 18 = 13
\]
This is true, so \((-1, -9)\) satisfies the first equation.

#### Equation 2: \(4x + 3y = 15\)
Substitute \(x = -1\) and \(y = -9\):
\[
4(-1) + 3(-9) = -4 - 27 = -31
\]
This is not equal to 15, so \((-1, -9)\) does not satisfy the second equation.

### Conclusion for Part (b)
Since \((-1, -9)\) does not satisfy both equations, **the ordered pair \((-1, -9)\) is not a solution to the system.**

Would you like a breakdown of each step in detail or have further questions?

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Here are some related questions to expand on this topic:

1. How do you verify solutions for systems of equations with three variables?
2. What are alternative methods to check if a point is a solution (e.g., graphing)?
3. How can you solve a system if given only one valid solution?
4. What strategies are there for solving systems where one equation is nonlinear?
5. How does changing coefficients in an equation affect its set of solutions?

**Tip:** To verify if a pair is a solution to a system, substitute it into each equation and ensure all equalities hold.

Determine if each ordered pair is a solution to the given system.
5x - 2y = 13
4x + 3y = 15
a. (3, 1)
b. (-1, -9)

Learn how to determine if ordered pairs are solutions to a system of linear equations. This step-by-step solution shows how to verify (3, 1) as a solution, and demonstrates why (-1, -9) is not. Suitable for students in Grades 8-10.

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