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Chapter 12: Problem 27

Solve each system. Use any method you wish. $$ \left\\{\begin{aligned} y &=3 x+2 \\ 3 x^{2}+y^{2} &=4 \end{aligned}\right. $$

Short Answer

Expert verified
The solutions are (0, 2) and (-1, -1).

Step by step solution

01

Substitute y from the First Equation

First, solve the first equation for y: \[ y = 3x + 2 \]Next, substitute this value of y into the second equation.
02

Form a Single Variable Equation

Substitute \( y = 3x + 2 \) into the second equation:\[ 3x^2 + (3x + 2)^2 = 4 \]Then, expand and simplify the equation.
03

Expand and Simplify

Expand \( (3x + 2)^2 \) to get:\[ 9x^2 + 12x + 4 \]Now the equation is:\[ 3x^2 + 9x^2 + 12x + 4 = 4 \]Simplify to get:\[ 12x^2 + 12x = 0 \]
04

Factor the Equation

Factor out the common term:\[ 12x(x + 1) = 0 \]This gives us two solutions for x:\[ x = 0 \textrm{ or } x = -1 \]
05

Find Corresponding y Values

Use the values of x to find y:For \( x = 0 \):\[ y = 3(0) + 2 = 2 \]For \( x = -1 \):\[ y = 3(-1) + 2 = -1 \]
06

Write the Solutions

We have two solutions:\[ (0, 2) \] and \[ (-1, -1) \]

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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 technique for solving systems of equations. It involves solving one equation for one variable and then substituting this expression into the other equation. In our example, we solved the first equation for y:
\( y = 3x + 2 \)
Next, we replaced y in the second equation. This transformed our system of equations into a single equation with one variable. Substitution simplifies complex systems, making them easier to solve step by step. It's particularly useful when one of the equations is already solved for a variable.
Quadratic Equations
Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). In our example, we ended up with a quadratic equation after substitution: \[ 12x^2 + 12x = 0 \].
This type of equation can be solved using several methods, including factoring, the quadratic formula, and completing the square. Quadratics often yield two solutions because they represent parabolic curves that can intersect the x-axis at two points.
Factoring
Factoring is a method used to solve quadratic equations by expressing the equation as a product of its factors. In our example, we factored the equation \[ 12x^2 + 12x = 0 \] by extracting the common factor: \[ 12x(x + 1) = 0 \].
This gave us two simpler equations to solve: \( 12x = 0 \) and \( x + 1 = 0 \). Factoring is a useful technique because it can quickly identify the solutions of the equation and requires only basic algebraic manipulation.

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Most popular questions from this chapter

Investments Kelly has \(\$ 20,000\) to invest. As her financial planner, you recommend that she diversify into three investments: Treasury bills that yield \(5 \%\) simple interest, Treasury bonds that yield \(7 \%\) simple interest, and corporate bonds that yield \(10 \%\) simple interest. Kelly wishes to earn \(\$ 1390\) per year in income. Also, Kelly wants her investment in Treasury bills to be \(\$ 3000\) more than her investment in corporate bonds. How much money should Kelly place in each investment?

Theater Revenues A Broadway theater has 500 seats, divided into orchestra, main, and balcony seating. Orchestra seats sell for \(\$ 150,\) main seats for \(\$ 135,\) and balcony seats for \(\$ 110 .\) If all the seats are sold, the gross revenue to the theater is \(\$ 64,250\). If all the main and balcony seats are sold, but only half the orchestra seats are sold, the gross revenue is \(\$ 56,750 .\) How many of each kind of seat are there?

Verify that the values of the variables listed are solutions of the system of equations. $$ \begin{array}{l} \left\\{\begin{array}{l} 4 x-z=7 \\ 8 x+5 y-z=0 \\ -x-y+5 z=6 \\ \end{array}\right.\\\ x=2, y=-3, z=1 \\ (2,-3,1) \end{array} $$

A doctor's prescription calls for the creation of pills that contain 12 units of vitamin \(\mathrm{B}_{12}\) and 12 units of vitamin E. Your pharmacy stocks three powders that can be used to make these pills: one contains \(20 \%\) vitamin \(\mathrm{B}_{12}\) and \(30 \%\) vitamin \(\mathrm{E} ;\) a second, \(40 \%\) vitamin \(\mathrm{B}_{12}\) and \(20 \%\) vitamin \(\mathrm{E}\) and a third, \(30 \%\) vitamin \(\mathrm{B}_{12}\) and \(40 \%\) vitamin \(\mathrm{E}\). Create \(\mathrm{a}\) table showing the possible combinations of these powders that could be mixed in each pill. Hint: 10 units of the first powder contains \(10 \cdot 0.2=2\) units of vitamin \(\mathrm{B}_{12}\).

Solve each system of equations. If the system has no solution, state that it is inconsistent. $$ \left\\{\begin{array}{l} 3 x-y=7 \\ 9 x-3 y=21 \end{array}\right. $$
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