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Chapter 4: Problem 42

In the following exercises, find three solutions to each linear equation. \(2 x+y=3\)

Short Answer

Expert verified
Three solutions are (0, 3), (1, 1), and (-1, 5).

Step by step solution

01

Rewrite the Equation

Start by rewriting the equation in terms of one variable. Solve for y in terms of x: \[ y = 3 - 2x \]
02

Choose a Value for x

Select a value for x. For instance, let x = 0. Substitute x = 0 into the equation: \[ y = 3 - 2(0) = 3 \]So, one solution is (0, 3).
03

Choose Another Value for x

Select another value for x. For instance, let x = 1. Substitute x = 1 into the equation: \[ y = 3 - 2(1) = 3 - 2 = 1 \]So, another solution is (1, 1).
04

Choose a Third Value for x

Select a third value for x. For instance, let x = -1. Substitute x = -1 into the equation: \[ y = 3 - 2(-1) = 3 + 2 = 5 \]So, a third solution is (-1, 5).

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Key Concepts

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

solving linear equations
Linear equations are equations of the first degree, meaning they have variables to the power of one. The general form is \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. To solve linear equations, we aim to find values for the variables that make the equation true.
Let's take the equation from the exercise: \(2x + y = 3\). First, we solve for one of the variables in terms of the other. Here, we choose to solve for \(y\):
\[ y = 3 - 2x \]
This transformation lets us find \(y\) if we know \(x\).
coordinate pairs
Coordinate pairs (also known as ordered pairs) are a pair of numbers that show the position of a point on a Cartesian plane. The pair is written in the form \( (x, y) \), where \(x\) is the horizontal coordinate and \(y\) is the vertical coordinate.
In the exercise, we found several solutions for \(2x + y = 3\) by choosing different values for \(x\) and calculating the corresponding \(y\)-values:
  • For \(x = 0\), \(y = 3\). The coordinate pair is \( (0, 3) \).
  • For \(x = 1\), \(y = 1\). The coordinate pair is \( (1, 1) \).
  • For \(x = -1\), \(y = 5\). The coordinate pair is \( (-1, 5) \).

These pairs show where the solutions lay on the graph of the equation.
substitution method
The substitution method involves solving one of the variables in terms of the other and then substituting back into the original equation. This method helps when dealing with systems of linear equations, but it鈥檚 also useful for finding solutions for single linear equations.
Let's illustrate this with the original problem of \(2x + y = 3\). First, we solved for \(y\):
\[ y = 3 - 2x \]
Then, we chose values for \(x\) and substituted them back into the new equation to find corresponding \(y\)-values:
  • Substituting \(x = 0\): \[y = 3 - 2(0) = 3 \]
  • Substituting \(x = 1\): \[y = 3 - 2(1) = 1 \]
  • Substituting \(x = -1\): \[y = 3 - 2(-1) = 5 \]

This method allows us to find precise solutions efficiently by concentrating on one variable at a time.

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Most popular questions from this chapter

Find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope-intercept form. . line \(4 x+5 y=-3\), point (0,0)

Graph the linear inequality \(y \geq 2\)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (3,-4) and (5,-4)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (0,4) and (2,-3)

Find the equation of each line. Write the equation in slope-intercept form. \(m=-\frac{3}{4}\), containing point (8,-5)
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