/*! 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 10 Graph each equation of the syste... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 10

Graph each equation of the system. Then solve the system to find the points of intersection. $$ \left\\{\begin{array}{l} y=\sqrt{x} \\ y=6-x \end{array}\right. $$

Short Answer

Expert verified
The points of intersection are (9, -3) and (4, 2).

Step by step solution

01

Graph the First Equation: y = sqrt(x)

Sketch the graph of the equation \( y = \sqrt{x} \). Start with some key points: when \( x = 0 \), \( y = 0 \); when \( x = 1 \), \( y = 1 \); when \( x = 4 \), \( y = 2 \), and so on. Plot these points on a coordinate plane and draw a curve passing through them.
02

Graph the Second Equation: y = 6 - x

Sketch the graph of the equation \( y = 6 - x \). This is a straight line. Calculate a few points by substituting values for \( x \). For instance: when \( x = 0 \), \( y = 6 \); when \( x = 6 \), \( y = 0 \). Plot these points and then draw the line.
03

Find the Points of Intersection

To find the points of intersection, set the two equations equal to each other: \( \sqrt{x} = 6 - x \). Square both sides to get rid of the square root: \(x = (6 - x)^2\). Expand and simplify it to find the values of \( x \).
04

Solve the Equation

Expand \( (6 - x)^2 \) to get \( x = 36 - 12x + x^2 \). Rearrange to form a quadratic equation: \( x^2 - 13x + 36 = 0 \). Solve this quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = -13 \), and \( c = 36 \).
05

Find the Roots

Using the quadratic formula, calculate the roots: \( x = \frac{13 \pm \sqrt{169 - 144}}{2} \), which simplifies to \( x = \frac{13 \pm 5}{2} \). The solutions are \( x = 9 \) and \( x = 4 \).
06

Find the Corresponding y-values

Substitute back into either original equation to find the corresponding \( y \)-values. For \( x = 9 \), \( y = 6 - 9 = -3 \); For \( x = 4 \), \( y = \sqrt{4} = 2 \). Thus, the points of intersection are \( (9, -3) \) and \( (4, 2) \).

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!

Key Concepts

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

graphing equations
Graphing equations is a fundamental skill in math that helps visualize solutions. When you graph an equation, you plot points on a coordinate plane. Each point \(x, y\) satisfies the equation.
For example, to graph \(y = \sqrt{x}\), calculate values for \(y\) using different \(x\) values. When \(x = 0\), \(y = 0\); when \(x = 1\), \(y = 1\); when \(x = 4\), \(y = 2\).
Plot these on a graph and join them with a smooth curve.
Likewise, for a line like \(y = 6 - x\), pick \(x\)-values and compute \(y\): When \(x = 0\), \(y = 6\); when \(x = 6\), \(y = 0\).
Connect these points with a straight line.
intersection points
Intersection points are where two graphs meet. These points represent solutions common to both equations.
To find them, set the equations equal. For \(y = \sqrt{x}\) and \(y = 6 - x\), set \(\sqrt{x} = 6 - x\).
Simplify and solve for \(x\): Square both sides to get \(x = (6 - x)^2\).
Expand to \(x = 36 - 12x + x^2\) and rearrange to form a quadratic equation: \(x^2 - 13x + 36 = 0\).
Solving this quadratic gives the \(x\)-values of the intersection points.
quadratic equations
Quadratic equations involve terms with \(x^2\). They often appear when working with parabolas.
The standard form is \(ax^2 + bx + c = 0\). To solve, use the quadratic formula: \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
For \(x^2 - 13x + 36 = 0\), set \(a = 1\), \(b = -13\), and \(c = 36\).
Substitute into the formula to get \( \frac{13 \pm \sqrt{169 - 144}}{2} \), simplifying to \( \frac{13 \pm 5}{2} \).
Solving gives roots \(x = 9\) and \(x = 4\).
These are the \(x\)-coordinates for intersection points.
square root
The square root operation finds a number which, when multiplied by itself, equals the original number.
It鈥檚 indicated by \sqrt{}\.
For example, \( \sqrt{9} = 3 \) because \( 3 * 3 = 9 \. \)
In the equation \(y = \sqrt{x}\), \(y\) is the positive root of \(x\). To graph, calculate points: \( \)like for \( x = 4 \, y = \sqrt{4} = 2 \).
Solving an equation with square roots often involves squaring both sides, but introduces extra solutions needing verification.
Like in \( \sqrt{x} = 6 - x \), squaring gives \( x = (6 - x)^2 \).

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 each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{r} x+y-z=6 \\ 3 x-2 y+z=-5 \\ x+3 y-2 z=14 \end{array}\right. $$

Computing Wind Speed The average airspeed of a single-engine aircraft is 150 miles per hour. If the aircraft flew the same distance in 2 hours with the wind as it flew in 3 hours against the wind, what was the wind speed?

Financial Planning A recently retired couple needs \(\$ 12,000\) per year to supplement their Social Security. They have \(\$ 300,000\) to invest to obtain this income. They have decided on two investment options: AA bonds yielding \(5 \%\) per annum and a Bank Certificate yielding \(2.5 \%\). (a) How much should be invested in each to realize exactly \(\$ 12,000 ?\) (b) If, after 2 years, the couple requires \(\$ 14,000\) per year in income, how should they reallocate their investment to achieve the new amount?

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Graph: \(f(x)=\frac{2 x^{2}-x-1}{x^{2}+2 x+1}\)

Determine whether the product is defined. If it is defined, find the product; if it is not write "not defined." $$ \left[\begin{array}{rrr} 4 & -2 & 3 \\ 0 & 1 & 2 \\ -1 & 0 & 1 \end{array}\right]\left[\begin{array}{rr} 2 & 6 \\ 1 & -1 \\ 0 & 2 \end{array}\right] $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Decision Maths

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.