/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 22 Solve each system. $$\begin{al... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 22

Solve each system. $$\begin{aligned} &x^{2}+y^{2}=49\\\ &x-2 y^{2}=7 \end{aligned}$$

Short Answer

Expert verified
The solution to the system of equations \(\begin{aligned} &x^{2}+y^{2}=49\\ &x-2 y^{2}=7 \end{aligned}\) is the ordered pair \((7, 0)\).

Step by step solution

01

Isolate a variable from either equation

We can isolate x from the second equation to get x in terms of y: \(x = 2y^2 + 7\)
02

Substitute the expression obtained in Step 1 into the first equation

Replace x in the first equation (circle) with the expression found in Step 1: \((2y^2 + 7)^2 + y^2 = 49\)
03

Simplify and solve the second equation for y

Expand and simplify the equation: \(4y^4 + 28y^2 + 49 + y^2 = 49\) Combine like terms: \(4y^4 + 29y^2 = 0\) Factor out the common factor of y^2: \(y^2(4y^2 + 29) = 0\) Solve for y: \(y^2 = 0\) or \(4y^2 + 29 = 0\) We find that \(y = 0\) or \(y^2 = -\dfrac{29}{4}\). Since y^2 cannot be negative, we discard the second root and determine that the only solution for y is \(y = 0\).
04

Find the value of x using the expression obtained in Step 1

Substitute y = 0 into the expression for x: \(x = 2(0)^2 + 7\) Therefore, \(x = 7\).
05

Write the solution as an ordered pair

Since we found that x = 7 and y = 0, the solution to this system of equations is the ordered pair \((7, 0)\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve using any method. Given that \(a=\log _{8} 225\) and \(b=\log _{2} 15,\) express \(a\) as a function of \(b\)

Let \(u\) represent \(1 / x, v\) represent \(1 / y,\) and \(w\) represent \(1 / z .\) Solve first for \(u, v,\) and \(w .\) Then solve the following system of equations: $$\begin{aligned} &\frac{2}{x}+\frac{2}{y}-\frac{3}{z}=3\\\ &\frac{1}{x}-\frac{2}{y}-\frac{3}{z}=9\\\ &\frac{7}{x}-\frac{2}{y}+\frac{9}{z}=-39 \end{aligned}$$

Solve the system of equations. $$\begin{aligned} w+x-y+z &=0 \\ -w+2 x+2 y+z &=5 \\ -w+3 x+y-z &=-4 \\ -2 w+x+y-3 z &=-7 \end{aligned}$$

Solve using any method. Given that \(f(x)=e^{x}-e^{-x},\) find \(f^{-1}(x)\) if it exists.

Use a graphing calculator to find the approximate solutions of the equation. $$\log _{5}(x+7)-\log _{5}(2 x-3)=1$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.