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Chapter 12: Problem 64

$$\text {Graph each equation.}$$ $$x^{2}-y^{2}=0$$

Short Answer

Expert verified
The graph forms two intersecting lines: y = x and y = -x.

Step by step solution

01

Identify the type of equation

The given equation is: \(x^2 - y^2 = 0\). This is a difference of squares equation which can be factored.
02

Factor the equation

Factor the equation using the difference of squares formula: \(a^2 - b^2 = (a + b)(a - b)\). So, for \(x^2 - y^2\), it becomes \((x + y)(x - y) = 0\).
03

Solve for the factors

Set each factor to zero to solve for the equations: \(x + y = 0\) and \(x - y = 0\). This gives us two linear equations: \(y = -x\) and \(y = x\).
04

Plot the equations

Plot the equations \(y = x\) and \(y = -x\) on the coordinate plane. These are straight lines passing through the origin (0,0) with slopes of 1 and -1 respectively.
05

Identify the graph

The graph of the equation \(x^2 - y^2 = 0\) consists of the two lines \(y = x\) and \(y = -x\). They form an 'X' shape on the coordinate plane.

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Key Concepts

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

Difference of Squares
The difference of squares is a common algebraic form that appears as \(a^2 - b^2\). This expression can be factored into \((a + b)(a - b)\), which simplifies solving and graphing.
The given equation, \(x^2 - y^2 = 0\), fits this form. By recognizing it as \(a^2 - b^2\), we can factor it out: \(x^2 - y^2 = (x + y)(x - y) = 0\). This factorization reveals two simple linear equations, \(x + y = 0\) and \(x - y = 0\), by setting each factor to zero.
The difference of squares simplifies complex quadratic equations into easy-to-graph linear equations. This method is crucial in algebra for solving and graphing equations efficiently. It's a powerful and essential concept to master.
Linear Equations
Linear equations are equations of the form \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. In our exercise, after factoring \(x^2 - y^2\), we get two linear equations: \(y = x\) and \(y = -x\).
Both these equations have slopes of \(1\) and \(-1\) respectively, and they intersect the origin, (0, 0). These straight lines will continue infinitely in both directions within the coordinate plane.
Here are some key properties:
  • These lines are symmetrical about the origin.
  • Their absolute value slopes are equal, but with opposite signs.
  • Since they both cross the origin, that point is a crucial intersection.
Recognizing and graphing linear equations is fundamental. Using simple forms like \(y = mx + c\) helps us understand and visualize relationships between variables.
Coordinate Plane
The coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane is represented by a pair of numbers, (x, y).
When graphing the equations \(y = x\) and \(y = -x\), we use the coordinate plane to plot each line. The origin, (0,0), is where the x-axis and y-axis intersect and serves as the reference point.
Steps to plot on the coordinate plane:
  • Identify the coordinates of key points, like the origin.
  • Plot each equation by determining where the line crosses both the x and y axes, if relevant.
  • Draw the lines ensuring they extend infinitely in both directions.
The graph of \(x^2 - y^2 = 0\) results in an 'X' shape forming from the intersections of \(y = x\) and \(y = -x\). This shape clearly illustrates the solution of the original algebraic equation on the coordinate plane.

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Most popular questions from this chapter

Use graphs to find the solution set to the system of inequalities: $$\begin{aligned}&y>2 x^{2}-3 x+1\\\&y<-2 x^{2}-8 x-1\end{aligned}$$

Determine whether the graph of each equation is a circle, parabola, ellipse, or hyperbola. $$ x^{2}-\frac{y^{2}}{9}=1 $$

Solve each problem. Find all points of intersection of the parabola \(y=x^{2}-\) \(2 x-3\) and the \(x\) -axis.

Sketch the graph of each ellipse. $$ \frac{(x-3)^{2}}{4}+\frac{(y-1)^{2}}{9}=1 $$

Graph each hyperbola and write the equations of its asymptotes. $$ x^{2}-\frac{y^{2}}{9}=1 $$
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