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Chapter 4: Problem 38

In the following exercises, find three solutions to each linear equation. \(x+y=6\)

Short Answer

Expert verified
Three solutions are (0, 6), (2, 4), and (4, 2).

Step by step solution

01

Choose three different values for x

To find three solutions to the equation, start by choosing three different values for the variable x. Let's choose x = 0, x = 2, and x = 4.
02

Substitute x = 0 into the equation

Substitute the chosen value x = 0 into the equation. Substitute: \(0 + y = 6\)Solve for y:\(y = 6\)First solution: (0, 6)
03

Substitute x = 2 into the equation

Substitute the chosen value x = 2 into the equation. Substitute: \(2 + y = 6\)Solve for y:\(y = 4\)Second solution: (2, 4)
04

Substitute x = 4 into the equation

Substitute the chosen value x = 4 into the equation. Substitute: \(4 + y = 6\)Solve for y:\(y = 2\)Third solution: (4, 2)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
A **linear equation** is an equation that models a relationship between two variables with a straight line when plotted on a graph. The general form of a linear equation in two variables is given by:
Ax + By = C
where A, B, and C are constants, and x and y are variables. To solve a linear equation, you need to find the values of the variables that make both sides of the equation equal.
In the exercise provided, the equation is:
x + y = 6
To find solutions to this equation, you can choose arbitrary values for one variable, solve for the other, and form a coordinate pair as your solution.
Algebraic Substitution
The process of **substitution** involves replacing one variable with a chosen value to solve for another variable. In our exercise, we used algebraic substitution to find values of y for different x values:
  • First, choose values for x. Here, we chose x = 0, 2, and 4.

  • Substitute each x value into the equation x + y = 6.

  • Solve for y by isolating it on one side of the equation.

Let's break it down with an example:
For x = 0:
Substitute: 0 + y = 6 (from the original equation)
Solve: y = 6
Thus, one solution is (0, 6). This process is repeated for x = 2 and x = 4 to find more solutions.
Coordinate Pairs
**Coordinate pairs** are sets of two numbers that represent a point on a Cartesian coordinate system. Each pair is written as (x, y) and shows the location of the point relative to the x and y axes. In the given exercise, we found three coordinate pairs:
  • (0, 6)

  • (2, 4)

  • (4, 2)

Each pair represents a solution to the equation x + y = 6. When you plot these points on a graph, they lie on the line that represents the equation. This helps visualize how the values of x and y relate to each other and verifies that they are indeed solutions to the equation.

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Most popular questions from this chapter

Explain why in some graphs of linear inequalities the boundary line is solid but in other graphs it is dashed.

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