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Chapter 4: Problem 186

Graph using the intercepts. \(x+y=-1\)

Short Answer

Expert verified
The intercepts are \((-1, 0)\) and \((0, -1)\), and the line passes through these points.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. Starting with the equation: \(x + y = -1\)Set \(y = 0\): \[ x + 0 = -1 \] Therefore, \[ x = -1 \]The x-intercept is at \((-1, 0)\).
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. Starting with the equation: \(x + y = -1\)Set \(x = 0\): \[ 0 + y = -1 \] Therefore, \[ y = -1 \]The y-intercept is at \((0, -1)\).
03

- Plot the intercepts on the graph

Plot the points \((-1, 0)\) and \((0, -1)\) on the graph. The x-intercept is at \((-1, 0)\), and the y-intercept is at \((0, -1)\).
04

- Draw the line

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

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Key Concepts

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

intercepts
When graphing linear equations, intercepts are very important. They are the points where the line crosses the axes. There are two kinds of intercepts: the x-intercept and the y-intercept.

Finding these intercepts helps in easily drawing the graph because they provide specific points to start from. You simply plot these points and draw the line through them.

Let's dive deeper into understanding each type.
x-intercept
The x-intercept is the point where the line crosses the x-axis. To find it, we set y to 0 in the equation and solve for x. This is because, at the x-intercept, the value of y is always 0.

Given the equation: \[x + y = -1\]

Set \[y = 0\]: \[x + 0 = -1\]
\[x = -1\]

So, the x-intercept for this equation is \((-1, 0)\). This means that the line crosses the x-axis at the point where x is -1.

Graphing this point involves locating -1 on the x-axis and marking it.
y-intercept
The y-intercept is the point where the line crosses the y-axis. To find it, we set x to 0 in the equation and solve for y. This is because, at the y-intercept, the value of x is always 0.

Given the equation: \[x + y = -1\]

Set \[x = 0\]: \[0 + y = -1\]
\[y = -1\]

So, the y-intercept for this equation is \((0, -1)\). This means that the line crosses the y-axis at the point where y is -1.

Graphing this point involves locating -1 on the y-axis and marking it.
plotting points
Now that we have our intercepts, we need to plot them on a graph. Marking these points helps us draw the line that represents the equation.

We have:
  • x-intercept: \((-1, 0)\)
  • y-intercept: \((0, -1)\)


To plot \((-1, 0)\), find -1 on the x-axis and plot the point where the y-coordinate is 0.

For \((0, -1)\), find -1 on the y-axis and plot the point where the x-coordinate is 0.

Once both points are plotted, draw a straight line through them. This line is the graph of the equation \[x + y = -1\].

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Most popular questions from this chapter

Find the equation of each line. Write the equation in slope-intercept form. Perpendicular to the line \(y\) -axis, point (-6,2)

553 . Shopping. Tula has $$\$ 20$$ to spend at the used book sale. Hardcover books cost $$\$ 2$$ each and paperback books cost $$\$ 0.50$$ each. If \(x\) is the number of hardcover books Tula can buy and \(y\) is the number of paperback books she can buy, the inequality \(2 x+\frac{1}{2} y \leq 20\) models the situation. (a) Graph the inequality. (b) List three solutions to the inequality \(2 x+\frac{1}{2} y \leq 20\) where both \(x\) and \(y\) are whole numbers.

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