Problem 87 - Introductory and Intermediate Algebra for College Students 4th

Question

Will help you prepare for the material covered in the first section of the next chapter. Is \((4,-1)\) a solution of both \(x+2 y=2\) and \(x-2 y=6 ?\)

Step-by-Step Solution

Answer

Yes, the point \((4,-1)\) is a solution to both equations.

Step 1: Substitute in first equation

In the first equation \(x+2y=2\), replace \(x\) with 4 and \(y\) with -1 to get: \(4 + 2(-1) = 4 - 2 = 2\). Since the left hand side equals the right hand side, the point is indeed on the line defined by the first equation.

Step 2: Substitute in second equation

In the second equation \(x-2y=6\), replace \(x\) with 4 and \(y\) with -1 to get: \(4 - 2(-1) = 4 + 2 =6\). Since the left hand side equals the right hand side, the point is indeed on the line defined by the second equation.

Key Concepts

Algebraic Solution Checking

When working with systems of linear equations, algebraic solution checking is a critical step to verify if a particular set of values—for example, an ordered pair like \( (4,-1) \)—satisfies both equations. To check a solution algebraically, each value from the pair is substituted into the respective variables of the equations.

For instance, in the equations \( x+2y=2 \) and \( x-2y=6 \) we substitute \( x \) with 4 and \( y \) with -1. If, after simplifying both sides of the equation, the resulting expressions are equal, the pair is a solution to that equation. Doing this for both equations in the system and obtaining equivalence confirms that the ordered pair is a solution to the system. This process illuminates the interplay between the equations, underscoring how their relationship governs the solution set.

Substitution Method

The substitution method is a technique used to solve systems of linear equations. The goal is to express one variable in terms of the other and then substitute this expression into the second equation. This helps to find the value of one variable and, subsequently, determine the value of the other.

However, in the context of checking solutions—not solving—substitution involves placing the values of an ordered pair into each equation to verify if the equations hold true. If the resulting expressions satisfy both equations, we have affirmed that the pair is indeed a solution to the system. This approach is powerful for verifying potential solutions, and provides a clear and systematic way to test the validity of an ordered pair. It is crucial to perform each step correctly to avoid errors that might lead to incorrect conclusions about a solution's validity.

Ordered Pairs

An ordered pair typically represents a point on the Cartesian coordinate system, where the first number in the pair is the x-coordinate and the second is the y-coordinate (in the form of \( (x,y) \) ). In the context of systems of linear equations, an ordered pair is a potential solution; it signifies where the graphs of the equations would intersect.

When checking whether an ordered pair is a solution to a system of equations, the pair is correctly tested by substitution into both equations. If the pair satisfies all the equations, it sits at the intersection of those lines on a graph. Therefore, understanding ordered pairs and their significance is vital in visualizing the solutions to systems of linear equations and thereby enhancing comprehension of how linear relationships are portrayed graphically.