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Chapter 5: Problem 64

Determine the number of solutions to the system of equations. $$ \begin{aligned} x^{2}-y^{2} &=0 \\ |x| &=|y| \end{aligned} $$

Short Answer

Expert verified
There are infinitely many solutions.

Step by step solution

01

Understand the system of equations

The given system has two equations: 1. The first equation is a difference of squares: \[ x^2 - y^2 = 0 \] 2. The second equation involves absolute values: \[ |x| = |y| \]
02

Factorize the first equation

Rewrite the first equation using the difference of squares formula: \[ x^2 - y^2 = (x + y)(x - y) = 0 \] This equation is satisfied when either \( x + y = 0 \) or \( x - y = 0 \).
03

Solve each factor separately

1. For \( x + y = 0 \), solve for one variable in terms of the other: \[ x = -y \]2. For \( x - y = 0 \), solve for one variable in terms of the other: \[ x = y \]
04

Consider the absolute value condition

From the second equation \( |x| = |y| \), there are two scenarios to consider for each case:1. If \( x = |y| \), then \( x = y \) or \( x = -y \).2. These conditions must match the solutions obtained from the first equation.
05

Compile valid solutions

- From \( x = y \) and considering the absolute value condition, possible solutions are \((x, y) = (a, a)\) for any real number 'a'.- From \( x = -y \) and considering the absolute value condition, possible solutions are \((x, y) = (a, -a)\) for any real number 'a'.
06

Count the number of solutions

Since there are no further restrictions, there are infinitely many solutions for each case (both \( x = y \) and \( x = -y \)), leading to infinitely many solutions in total.

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Key Concepts

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

Difference of Squares
The concept of the 'difference of squares' is a fundamental algebraic technique. It鈥檚 important because it helps us simplify quadratic expressions and solve equations. The difference of squares refers to expressions of the form \[ a^2 - b^2 \]. The neat trick here is that it can be factored into a product of two binomials: \[ a^2 - b^2 = (a + b)(a - b) \]. This form is easier to work with, especially for solving equations.

Let's look at the example from the exercise:
  • The first equation we have is \[ x^2 - y^2 = 0 \].
  • Applying the difference of squares formula, we rewrite it as \[ (x + y)(x - y) = 0 \].
  • This tells us that either \[ x + y = 0 \] or \[ x - y = 0 \] or both.
This factorization is critical because it breaks down a complex equation into simpler ones. Once factored, it鈥檚 just a matter of solving linear equations!
Absolute Value Equations
Another important aspect of our exercise is 'absolute value equations'. The absolute value \(|x|\) of a number is its distance from zero on the number line, regardless of direction. So, \(|x| = a\) means \(x\) is either \(a\) or \(-a\).

Consider the second equation in the problem:
  • \(|x| = |y|\) tells us that the absolute values of \(x\) and \(y\) are the same.
  • This implies two cases: \(x = y\) or \(x = -y\).
Handling absolute values, we always need to think of both positive and negative scenarios. This ensures we don鈥檛 miss any solutions<.
Solutions to Equations
Finally, let's discuss 'solutions to equations'. Solving equations is about finding all values that make the equation true. In this exercise:
  • We derived \(x = y\) and \(x = -y\) from our factored form and absolute value conditions.
  • We then matched these with the solutions from the difference of squares.
  • Upon analysis, it鈥檚 clear that both \(x = y\) and \(x = -y\) offer infinitely many solutions.
So, in the end, any real number \(a\) gives us a solution, either as \( (a,a)\) or \((a, -a)\). That's why the system has infinitely many solutions. Understanding this gives us a deeper insight into the behavior of equations and their solutions.

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Most popular questions from this chapter

Solve the system of equations by using the substitution method. (See Example 2\()\) $$ \begin{array}{l} 3 x=2 y-11 \\ 6+5 x=1 \end{array} $$

Write a system of inequalities that represents the points in the first quadrant less than 3 units from the origin.

Use the given constraints to find the maximum value of the objective function and the ordered pair \((x, y)\) that produces the maximum value. \(x \geq 0, y \geq 0\) \(3 x+4 y \leq 48\) \(2 x+y \leq 22\) \(y \leq 9\) a. Maximize: \(z=100 x+120 y\) b. Maximize: \(z=100 x+140 y\)

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(y=-\frac{1}{5} x+2\) \(2 x+10 y=10\) a. (5,1) b. (-10,4)

a. Sketch the lines defined by \(y=2 x\) and \(y=-\frac{1}{2} x+5\) b. Find the area of the triangle bounded by the lines in part (a) and the \(x\) -axis.
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