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Chapter 2: Problem 45

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 2 x+y=0 $$

Short Answer

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

Step by step solution

01

Find the x-intercept

To find the x-intercept, set y to 0 in the equation and solve for x. So, from the equation, we get:\[2x + 0 = 0\]Thus:\[2x = 0\]Divide both sides by 2:\[x = 0\]Therefore, the x-intercept is (0, 0).
02

Find the y-intercept

To find the y-intercept, set x to 0 in the equation and solve for y. So, from the equation, we get:\[2(0) + y = 0\]Thus:\[y = 0\]Therefore, the y-intercept is (0, 0).
03

Graph the equation

Graph the equation by plotting the intercepts. In this case, both the x-intercept and y-intercept are the same point, (0, 0). Since this is a linear equation, draw a straight line through the point, which will be a line through the origin with a slope of -2.

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Key Concepts

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

linear equations
A linear equation is a mathematical expression that forms a straight line when graphed on a coordinate system. These equations are typically written in the form \( ax + by = c \) where \( a \), \( b \), and \( c \) are constants. The key features of a linear equation include:
  • A graph that is a straight line.
  • Constant rates of change or slope.
  • Variables with no exponents higher than one.
Linear equations are foundational in algebra and are used to model real-world situations where there is a constant rate of change. Understanding how to manipulate and graph these equations is crucial for solving problems and visualizing data.
graphing lines
Graphing lines involves plotting points on a coordinate plane and drawing a line through these points. For a linear equation, the most useful points to graph are the x- and y-intercepts. Here's a brief overview of the steps involved in graphing linear equations:
  • Identify Intercepts: Find where the line crosses the x-axis (x-intercept) and y-axis (y-intercept).
  • Plot Points: Place the intercepts on the coordinate plane.
  • Draw the Line: Connect the intercepts with a straight line, extending it across the plane.
In some cases, additional points might be needed to ensure accuracy. The line should represent all solutions to the equation. For example, for the equation \(2x + y = 0\), the line passes through both intercepts at (0,0) and extends according to its slope.
finding intercepts
Finding the intercepts of a linear equation is a critical step in graphing the line. The x-intercept is where the line crosses the x-axis, and the y-intercept is where it crosses the y-axis.
  • Finding the x-intercept: Set \(y\) to 0 and solve for \(x\). For our example equation \(2x + y = 0\), if \(y = 0\), you get \(2x = 0\), thus \(x = 0\).
  • Finding the y-intercept: Set \(x\) to 0 and solve for \(y\). Again for our example, if \(x = 0\), you get \(y = 0\).
Once intercepts are found, they can be graphed as points on the coordinate plane. These points help in drawing the straight line that represents the linear equation. Intercepts provide a direct insight into how the equation behaves at critical points on the axes.

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Most popular questions from this chapter

The average price of a movie ticket in 2004 was \(\$ 6.21 .\) In \(2016,\) the average price was \(\$ 8.65 .\) Find and interpret the average rate of change in the price of a movie ticket per year to the nearest cent.

An equation that defines \(y\) as a function \(f\) of \(x\) is given. (a) Solve for \(y\) in terms of \(x\), and write each equation using function notation \(f(x) .\) (b) Find \(f(3)\). $$ -2 x+5 y=9 $$

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

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ f=\\{(-1,3),(4,7),(0,6),(2,2)\\} $$

Use your knowledge of the slopes of parallel and perpendicular lines. Show that \((-13,-9),(-11,-1),(2,-2),\) and (4,6) are the vertices of a parallelogram. (Hint: A parallelogram is a four-sided figure with opposite sides parallel.)
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