Problem 88 - 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 \((-4,3)\) a solution of both \(x+2 y=2\) and \(x-2 y=6 ?\)

Step-by-Step Solution

Answer

No, the point (-4,3) is not a solution to both equations.

Step 1: Check the first equation

Substitute the values from the point into the first equation: \((-4) + 2*3 = -4 + 6 = 2\). The left side equals the right side, so the point (-4,3) satisfies the first equation.

Step 2: Check the second equation

Substitute the values from the point into the second equation: \((-4) - 2*3 = -4 - 6 = -10\). The left side does not equal the right side, so the point (-4,3) does not satisfy the second equation.

Step 3: Conclude

Since the point (-4,3) only satisfies one of the two given equations, it is not a solution to both equations.

Key Concepts

Understanding Coordinates

Coordinates are a way to locate points on a plane using an ordered pair (x, y). This "ordered pair" tells us where to position a point horizontally and vertically.
A simple analogy is thinking about aisles and shelves in a library, where

• "x" tells you which aisle to go to (left or right).
• "y" tells you which shelf to look up (up or down).

The point \((-4, 3)\) means you move 4 units left from the origin (0,0), and 3 units up. This helps us precisely find where a point lies on a coordinate grid. Knowing coordinates is crucial when working with equations because they help us test whether a particular point satisfies an equation.

Finding the Solution

A solution, in the context of equations, is a set of values that makes the equation true.
For a system of equations (like in our exercise), a solution would be a point that satisfies all the equations involved.
Here’s how to verify if a point is a solution:

• Substitute the x and y values from the coordinate into each equation.
• Solve to see if the left side equals the right side.

In our example, we substituted \((-4, 3)\) and found it only satisfied the first equation \(x + 2y = 2\) but not the second \(x - 2y = 6\).
This means \((-4, 3)\) isn't a solution to the system because it doesn't meet both conditions. A valid solution should work for every equation in the system.

Exploring Linear Equations

Linear equations are mathematical statements that make a straight line when graphed.
They are usually written in the form \(ax + by = c\), where x and y are variables, and a, b, and c are constants. Linear equations have:

• One solution where two lines intersect, known as the point of intersection.
• No solution if they are parallel and never meet.
• Infinitely many solutions if they are the same line.

In our problem, both equations represent straight lines on a graph.
When we tested the point \((-4, 3)\), it only sat on one line, meaning the lines intersect at a different point, or not at all, so \((-4, 3)\) isn't where they meet.
Understanding linear equations helps us see patterns and relationships between variables in everyday life.