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Chapter 7: Problem 25

For the following exercises, use any method to solve the nonlinear system. $$ \begin{array}{r} x^{2}-y^{2}=9 \\ x=3 \end{array} $$

Short Answer

Expert verified
The solution is \(x = 3\) and \(y = 0\).

Step by step solution

01

Substitute Known Value

We are given two equations: the first is a nonlinear equation, \(x^2 - y^2 = 9\), and the second is \(x = 3\). Substitute \(x = 3\) into the first equation to find the value of \(y\).
02

Calculate Expression

Replace \(x\) with 3 in the first equation, so it becomes \(3^2 - y^2 = 9\). This leads to \(9 - y^2 = 9\).
03

Simplify the Equation

Subtract 9 from both sides, resulting in \(-y^2 = 0\). Simplifying gives \(y^2 = 0\).
04

Solve for y

Take the square root of both sides to solve for \(y\). Since \(y^2 = 0\), we have \(y = 0\).
05

Verify the Solution

Verify the solution by plugging \(x = 3\) and \(y = 0\) back into the original equations. Check if \(3^2 - 0^2 = 9\), which simplifies to \(9 = 9\). This confirms the solution is correct.

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Key Concepts

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

Substitution Method
The substitution method is a powerful tool used to solve systems of equations, especially when one of the equations is straightforward to manipulate. To use the substitution method effectively, follow these basic steps:
  • Identify an easy substitution: Find an equation that you can simplify easily or already has a variable expressed in terms of another.
  • Substitute the expression: Replace the variable in the second equation with this expression. This helps reduce the number of variables in that equation, making it easier to solve.
In this exercise, the equation "\(x=3\)" immediately provides a value for \(x\). Simply substitute \(x=3\) into the nonlinear equation \(x^2-y^2=9\). This transforms the equation into "\(3^2-y^2=9\)", simplifying the problem significantly by reducing it to a single variable problem.
Simplifying Equations
Simplifying equations is an essential skill that helps in making complex problems more manageable. Once you've substituted a value, like in our step where \(x=3\) is plugged into \(x^2-y^2=9\), simplification follows.
  • Perform operations: Do simple arithmetic operations, such as addition, subtraction, multiplication, or division, to condense the equation.
  • Isolate variables: Work towards expressions that isolate variables on one side of the equation to directly solve for them.
In this exercise, substituting \(x=3\) yields an expression \(9-y^2=9\). Simplifying \(9-y^2=9\) further involves moving terms around to find \(y^2=0\). Simplification converts initially complex equations into straightforward solutions.
Verifying Solutions
Verification is the final and crucial step in solving any equation system. It confirms that the solutions satisfy the original equations.
  • Re-substitute solutions: After obtaining potential solutions for variables, substitute them back into the original equations.
  • Check all conditions: Ensure that all equations are satisfied with these solutions.
Here, we substitute \(x=3\) and \(y=0\) back into the original system: \(x^2-y^2=9\). Plugging these values, \(3^2-0^2=9\), verifies that the solution holds true, as \(9=9\). Successful verification confirms the correctness of your solution, ensuring you've solved the problem accurately.

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Most popular questions from this chapter

For the following exercises, write a system of equations that represents the situation. Then, solve the system using the inverse of a matrix. A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at \(\$ 1\) and the chocolate chip cookies at \(\$ 0.75\). They raised \(\$ 700\) and sold 850 items. How many brownies and how many cookies were sold?

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. You invest \(\$ 10,000\) into two accounts, which receive \(8 \%\) interest and \(5 \%\) interest. At the end of a year, you had \(\$ 10,710\) in your combined accounts. How much was invested in each account?

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 5 x-4 y=2 \\ -4 x+7 y=6 \end{array} $$

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer's Rule. At a women's prison down the road, the total number of inmates aged \(20-49\) totaled \(5,525 .\) This year, the \(20-29\) age group increased by \(10 \%,\) the 30-39 age group decreased by \(20 \%,\) and the \(40-49\) age group doubled. There are now 6,040 prisoners. Originally, there were 500 more in the \(30-39\) age group than the \(20-29\) age group. Determine the prison population for each age group last year.

For the following exercises, solve the system of linear equations using Cramer's Rule. $$ \begin{array}{r} 4 x-6 y+8 z=10 \\ -2 x+3 y-4 z=-5 \\ 12 x+18 y-24 z=-30 \end{array} $$
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