Problem 15 - Algebra 1

Question

Decide whether the ordered pair is a solution of the system of linear equations. $$ \begin{array}{ll} -15 x+7 y=1 & (3,5) \\ 3 x-y=1 \end{array} $$

Step-by-Step Solution

Answer

No, the ordered pair (3,5) is not a solution to the given system of linear equations.

Step 1: Substitution into first equation

Let's start from the first equation: -15x + 7y = 1. Here, for the ordered pair (3,5), \( x = 3 \) and \( y = 5 \). Substitute \( x \) and \( y \) into the equation: -15(3) + 7(5).

Step 2: Solve the first equation

Solving the equation we get -45 + 35 = -10 which is not equal to 1. So (3,5) isn't a solution to the first equation.

Step 3: Substitution into second equation

Now let's follow the same steps for the second equation: 3x - y = 1. Substitute \( x = 3 \) and \( y = 5 \): 3(3) - 5.

Step 4: Solve the second equation

Solving the equation we get 9 - 5 = 4 which is not equal to 1. So (3,5) isn't a solution to the second equation either.

Key Concepts

Ordered Pair

An ordered pair is a fundamental concept in mathematics, particularly in the context of systems of equations. It consists of two elements, usually denoted as

• \((x, y)\)
• \((a, b)\)

The order matters, meaning \((3, 5)\) is different from \((5, 3)\). In a coordinate plane, these values represent the

• horizontal (x) position
• vertical (y) position

Ordered pairs are essential for determining whether a particular point satisfies a system of equations. In our exercise, we're asked if the ordered pair \((3, 5)\) is a solution to a set of linear equations. To verify this, we must see if both parts of the pair make the equations true when substituted. If they do, the ordered pair is a solution of the system. Otherwise, it's not.

Substitution Method

The substitution method is a technique used to determine if a specific ordered pair is a solution for a given system of equations. Here's how it works:First, identify the values from the ordered pair. In our example, \(x = 3\) and \(y = 5\). Then, substitute these values into each equation of the system.Consider the equations:

• First equation: \(-15x + 7y = 1\)
• Second equation: \(3x - y = 1\)

Substitute \(x = 3\) and \(y = 5\) into these equations:For the first equation:
- Calculate \(-15(3) + 7(5)\)- This simplifies to \(-45 + 35 = -10\)- Since \(-10\) does not equal \(1\), the ordered pair \((3, 5)\) is not a solution.For the second equation:
- Calculate \(3(3) - 5\)- This simplifies to \(9 - 5 = 4\)- Since \(4\) does not equal \(1\), the ordered pair \((3, 5)\) again does not satisfy the equation.By following the substitution method step-by-step, we ensured we correctly evaluated the given ordered pair against each equation.

Solution Verification

Solution verification is a crucial step in solving systems of equations. It ensures that the ordered pair truly satisfies all equations in the system. Here’s a simple process to verify the solution:Once substitution is complete:

• Check the resulting values.
• Ensure each calculation aligns with the equation’s outputs.

If the computations accurately match the right side of each equation, the ordered pair is a verified solution.In our exercise:
After substituting \((3, 5)\) into both equations, the calculated results \(-10\) and \(4\) did not match the right side values \(1\). This mismatch shows \((3, 5)\) is not a solution.Verification helps avoid errors. It confirms the ordered pair genuinely works for the system, providing confidence in the solution's validity.