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Chapter 11: Problem 168

In the following exercises, graph using the intercepts. $$ 3 x+y=3 $$

Short Answer

Expert verified
The x-intercept is (1, 0) and the y-intercept is (0, 3).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. So, we plug in y = 0 into the equation:\(3x + y = 3\)This becomes:\(3x + 0 = 3\)Solve for x:\(3x = 3\)\(x = 1\)Therefore, the x-intercept is (1, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. So, we plug in x = 0 into the equation:\(3x + y = 3\)This becomes:\(3(0) + y = 3\)\(y = 3\)Therefore, the y-intercept is (0, 3).
03

Plot the intercepts on the graph

Plot the points (1, 0) for the x-intercept and (0, 3) for the y-intercept on the coordinate plane.
04

Draw the line

Draw a straight line through the points (1, 0) and (0, 3). This is the graph of the equation \(3x + y = 3\).

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Key Concepts

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

x-intercept
The x-intercept is the point where a graph crosses the x-axis. This happens when the y-value is zero. To find it, set y to 0 in your equation and solve for x.
For example, in the equation provided, we substitute y with 0:
3x + 0 = 3
Solving for x, we get:
3x = 3
Dividing both sides by 3, x = 1.
So, the x-intercept is at the point (1, 0).
Remember, the x-intercept always means that the y-coordinate is zero. This is crucial when graphing linear equations, as it gives you a specific point you can confidently plot on your graph.
y-intercept
The y-intercept is the point where a graph crosses the y-axis. In this case, the x-value is zero.
To find it, set x to 0 in your equation and solve for y.
Using the same equation: 3x + y = 3
Substitute x with 0:
3(0) + y = 3
This simplifies to:
y = 3
Therefore, the y-intercept is at the point (0, 3).
The y-intercept always means that the x-coordinate is zero. This helps you plot another key point on your graph, making it easier to draw a straight line to represent the equation.
coordinate plane
The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers: the x-coordinate and the y-coordinate.
The plane is divided by a horizontal line called the x-axis and a vertical line called the y-axis. These lines intersect at a point known as the origin (0,0).
When graphing linear equations, you plot points along these axes. Each point has an x-value and a y-value.
For example, the point (1, 0) has an x-value of 1 and a y-value of 0. Similarly, the point (0, 3) has an x-value of 0 and a y-value of 3.
By plotting these points, you can visualize the relationship described by your equation.
solving linear equations
Solving linear equations involves finding the values of x and y that satisfy the equation.
Typically, you follow these steps:
  • Identify the equation, such as 3x + y = 3.
  • Find the x-intercept by setting y to 0 and solving for x.
  • Find the y-intercept by setting x to 0 and solving for y.
  • Plot these intercepts on the coordinate plane.
  • Draw a straight line that passes through these points.
Linear equations form straight lines on the coordinate plane. Once you have the intercepts, drawing the line helps visualize the solutions. Every point on the line represents a pair (x, y) that makes the equation true.

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