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Chapter 2: Problem 43

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ x+5 y=0 $$

Short Answer

Expert verified
Both intercepts are at (0, 0), and the graph is a line through the origin.

Step by step solution

01

- Find the x-intercept

To find the x-intercept, set y to 0 and solve for x. The equation becomes\[ x + 5(0) = 0 \]which simplifies to\[ x = 0 \]Thus, the x-intercept is at the point (0, 0).
02

- Find the y-intercept

To find the y-intercept, set x to 0 and solve for y. The equation becomes\[ 0 + 5y = 0 \]which simplifies to\[ y = 0 \]Thus, the y-intercept is at the point (0, 0).
03

- Plot the intercepts

On a coordinate plane, plot the x-intercept (0, 0) and the y-intercept (0, 0). Since both intercepts are at the same point, this actually gives just one point on the plane.
04

- Draw the graph

Since the equation is linear and both intercepts are at (0, 0), the graph of the equation is a line that passes through the origin. Any additional points that satisfy the equation can also be plotted to ensure the line is correctly drawn.

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Key Concepts

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

X-Intercepts
To find the x-intercept of an equation, you need to set the value of y to 0 and then solve the equation for x.
This tells you where the line crosses the x-axis on the coordinate plane.
For example, in the equation given, setting y to 0 simplifies the equation to:
\[x + 5(0) = 0 \]
which further simplifies to: \[x = 0 \]
Therefore, the x-intercept is at the coordinate point (0, 0).
Each x-intercept is a point where the line meets the x-axis.
  • To find x-intercepts: Set y to 0.
  • Solve the resulting equation for x.
Remember, the x-intercept may sometimes correspond to the origin (0, 0).
Y-Intercepts
Just like finding the x-intercept, finding the y-intercept involves setting the other variable (x, in this case) to 0.
This way, you determine where the line crosses the y-axis.
For the given equation, setting x to 0 simplifies it to:
\[0 + 5y = 0 \]
which further simplifies to \[y = 0 \]
So, the y-intercept is the coordinate point (0, 0).
It is possible for the y-intercept and the x-intercept to both be at the origin, as in this example.
  • Find the y-intercept by setting x to 0.
  • Solve for y.
Each y-intercept is a point where the line meets the y-axis.
Coordinate Plane
A coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves defined by coordinates.
It features an x-axis (horizontal) and a y-axis (vertical), which intersect at the origin.
It is divided into four quadrants.
  • Quadrant I: Both x and y are positive.
  • Quadrant II: x is negative, y is positive.
  • Quadrant III: Both x and y are negative.
  • Quadrant IV: x is positive, y is negative.
When graphing equations, we plot intercepts and other points on this plane to visualize the relationship.
Origin
The Origin is the point where the x-axis and y-axis intersect on the coordinate plane.
It is denoted by the coordinates (0, 0).
In this specific example, both the x-intercept and y-intercept are at the origin.
The origin is a crucial reference point in graphing.
  • It represents the starting point (0,0).
  • It divides the plane into four quadrants.
The origin helps in understanding how lines and curves are positioned relative to each axis.

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Most popular questions from this chapter

Determine whether each pair of lines is parallel, perpendicular, or neither. \(4 x+y=0\) and \(5 x-8=2 y\)

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ \begin{array}{c|c} x & y=f(x) \\ \hline 2 & 4 \\ \hline 1 & 1 \\ \hline 0 & 0 \\ \hline-1 & 1 \\ \hline-2 & 4 \end{array} $$

Determine whether each pair of lines is parallel, perpendicular, or neither. \(2 x=y+3\) and \(2 y+x=3\)

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ \begin{array}{c|c} x & y=f(x) \\ \hline 8 & 6 \\ \hline 5 & 3 \\ \hline 2 & 0 \\ \hline-1 & -3 \\ \hline-4 & -6 \end{array} $$

Graph the intersection of each pair of inequalities. $$ x+y \leq 1 \quad \text { and } \quad x \geq 1 $$
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