Problem 41 - Algebra 1

Question

Which ordered pair is a solution of the linear system? $$ \begin{aligned} &x+y=0.5\\\ &x+2 y=1 \end{aligned} $$ \(\begin{array}{llll}\text { (A) }(0,-2) & \text { (B) }(-0.5,0) & \text { C } & (0,0.5)\end{array}\) (D) \((0,-0.5)\)

Step-by-Step Solution

Answer

The solution to the given linear system of equations is ordered pair (C) (0, 0.5).

Step 1: Substitute (A) into both equations

Substitute ordered pair (A) \( (0, -2) \) into the equations as follows: \n In the first equation, replace \(x\) with 0 and \(y\) with -2: \(0 - 2 = -2\), not equal to 0.5. Therefore, (0, -2) is not a solution to the system.

Step 2: Substitute (B) into both equations

Substitute ordered pair (B) \(-0.5,0\) into the equations. Replace \(x\) with -0.5 and \(y\) with 0 in both the equations: \n In the first equation, -0.5 + 0 = -0.5, which is not equal to 0.5. Therefore, (-0.5, 0) is not a solution to the system.

Step 3: Substitute (C) into both equations

Now, substitute ordered pair (C) (0, 0.5) into the equations. Replace \(x\) with 0 and \(y\) with 0.5: \n In the first equation, 0 + 0.5 = 0.5, and in the second equation, 0 + 2 * 0.5 = 1. Both equal the respective constants from the equations. Therefore, (0, 0.5) is a solution to the system.

Step 4: No need to substitute (D)

Given that multiple choice questions have only one correct answer and we have already found that (C) (0, 0.5) is a valid solution, there is no need to test the fourth option.

Key Concepts

Ordered Pairs

In linear algebra, an ordered pair is a pair of numbers used to locate a specific point on a coordinate plane. The order is crucial as it dictates which point is the x-coordinate and which is the y-coordinate. For example, in the ordered pair \((x, y)\), \(x\) is the horizontal coordinate (position on the x-axis), while \(y\) is the vertical coordinate (position on the y-axis).
Understanding ordered pairs is essential because they enable us to express solutions to equations and systems of equations. Essentially, when we find an ordered pair that satisfies both equations in a linear system, we have determined the point where their lines intersect.
Here are some key points about ordered pairs:

• The first number in an ordered pair is always the x-coordinate.
• The second number is always the y-coordinate.
• Ordered pairs can represent solutions to equations or systems.

Substitution Method

The substitution method is a popular technique for solving systems of linear equations. It involves substituting the expression of one variable from one equation into another equation. This substitution reduces the system to a single equation with one variable, simplifying the problem significantly.
Let's break it down:

• Start by solving one of the equations for one variable.
• Substitute the expression found into the other equation.
• Solve for the remaining variable.
• Substitute back to find the first variable.

Using the substitution method helps us explore solutions to the system step by step. It also provides a clear path to ensure we are finding consistent solutions that work for both equations.

Solutions of Equations

A solution to a system of equations is essentially the point where the equations intersect on a graph. In the context of linear equations, these solutions are often represented as ordered pairs \((x, y)\). Finding a solution means finding values for the variables that satisfy all equations in the system simultaneously.
For the given problem, we used the substitution method to identify that the ordered pair \((0, 0.5)\) is a solution. This ordered pair resolves both equations, satisfying them fully. Testing solutions systematically ensures that each potential solution fulfills each equation's requirements.
Here's a recap of the process for finding solutions:

• Test each ordered pair in the context of both equations.
• Ensure that substitution results in true statements for both equations.
• Confirm the ordered pair aligns with the intersection point of the equations' lines on a graph.

Approaching solutions in this manner ensures accuracy and confirms the validity of the potential answers for linear systems.