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Chapter 5: Problem 30

Solve the system by using any method. $$ \begin{array}{l} y=x^{2}-6 x+9 \\ y=-2 x+5 \end{array} $$

Short Answer

Expert verified
The solution is \((2, 1)\).

Step by step solution

01

Set the Equations Equal

Both equations are equal to y, so set the right-hand sides equal to each other: \[x^2 - 6x + 9 = -2x + 5\]
02

Rearrange the Equation

Rearrange the equation to bring all terms to one side: \[x^2 - 6x + 2x + 9 - 5 = 0\] Simplify to: \[x^2 - 4x + 4 = 0\]
03

Factor the Quadratic Equation

Factor \(x^2 - 4x + 4\) as: \[(x-2)^2 = 0\]
04

Solve for x

Solve the factored equation by finding the value of x: \[(x-2) = 0\] Thus, \[x = 2\]
05

Solve for y

Substitute \(x = 2\) back into either original equation to find y. Using \(y = -2x + 5\): \[y = -2(2) + 5 = -4 + 5 = 1\]
06

Solution

The solution to the system of equations is \((2, 1)\)

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Key Concepts

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

Quadratic Equations
Quadratic equations are polynomials of degree 2. They are usually written in the form \(ax^2 + bx + c = 0\). In the given problem, the equation \(y = x^2 - 6x + 9\) represents a quadratic equation where the terms \(a = 1\), \(b = -6\), and \(c = 9\). Quadratic equations can have up to two real solutions. They are graphically represented as parabolas, which can open upwards or downwards, depending on the sign of \(a\). Solving quadratic equations often involves methods such as factoring, completing the square, or using the quadratic formula.
Factoring
Factoring is a method of breaking down an equation into simpler components called factors. In the given exercise, we factor the quadratic equation \(x^2 - 4x + 4\). This equation can be factored as \((x - 2)^2\).This means that both roots of the quadratic equation are \(x = 2\). Factoring simplifies solving equations and allows us to identify the roots quickly. Here is a step-by-step approach to factoring a quadratic equation:
  • 1. Write down the quadratic equation.
  • 2. Rearrange the terms to group them for easy factoring.
  • 3. Identify binomial squares or common factors.
  • 4. Rewrite the equation in a factored form.
Substitution Method
The substitution method involves substituting one equation into another to solve a system of equations. In the given problem, we set both equations equal to each other because they both equal \(y\):\(x^2 - 6x + 9 = -2x + 5\).Here鈥檚 how the substitution method works in this example:
  • 1. Set the two equations equal to each other.
  • 2. Rearrange the equation to solve for one variable (e.g., \(x\)).
  • 3. Simplify and solve the resulting equation.
  • 4. Use the value of \(x\) to find \(y\) by substituting back into any original equation.
This method is especially useful when one of the equations is already solved for one of the variables.
Algebraic Manipulation
Algebraic manipulation involves using algebraic techniques to simplify and solve equations. In this problem, we use various algebraic manipulations to solve the system of equations:First, we set the right-hand sides of both equations equal:\(x^2 - 6x + 9 = -2x + 5\).Then, we rearrange to combine all terms on one side:\(x^2 - 4x + 4 = 0\).Next, we factor the quadratic equation and find the value of \(x\):\((x - 2)^2 = 0\) leading to \(x = 2\).Finally, we substitute \(x = 2\) back into the linear equation to solve for \(y\):\(y = -2(2) + 5 = 1\).Effective algebraic manipulation is key to solving various types of equations and systems of equations.

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Most popular questions from this chapter

Let \(x\) represent the number of country songs that Sierra puts on a playlist on her portable media player. Let \(y\) represent the number of rock songs that she puts on the playlist. For parts (a)-(e), write an inequality to represent the given statement. a. Sierra will put at least 6 country songs on the playlist. b. Sierra will put no more than 10 rock songs on the playlist. c. Sierra wants to limit the length of the playlist to at most 20 songs. d. The number of country songs cannot be negative. e. The number of rock songs cannot be negative. f. Graph the solution set to the system of inequalities from parts (a)-(e).

Solve the system using any method. $$ \begin{array}{l} y=-0.18 x+0.129 \\ y=-0.15 x+0.1275 \end{array} $$

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)

The attending physician in an emergency room treats an unconscious patient suspected of a drug overdose. The physician does not know the initial concentration \(A_{0}\) of the drug in the bloodstream at the time of injection. However, the physician knows that after \(3 \mathrm{hr}\), the drug concentration in the blood is \(0.69 \mu \mathrm{g} / \mathrm{dL}\) and after \(4 \mathrm{hr}\), the concentration is \(0.655 \mu \mathrm{g} / \mathrm{dL}\). The model \(A(t)=A_{0} e^{-k t}\) represents the drug concentration \(A(t)\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) in the bloodstream \(t\) hours after injection. The value of \(k\) is a constant related to the rate at which the drug is removed by the body. a. Substitute 0.69 for \(A(t)\) and 3 for \(t\) in the model and write the resulting equation. b. Substitute 0.655 for \(A(t)\) and 4 for \(t\) in the model and write the resulting equation. c. Use the system of equations from parts (a) and (b) to solve for \(k .\) Round to 3 decimal places. d. Use the system of equations from parts (a) and (b) to approximate the initial concentration \(A_{0}\) (in \(\mu \mathrm{g} / \mathrm{dL}\) ) at the time of injection. Round to 2 decimal places. e. Determine the concentration of the drug after \(12 \mathrm{hr}\). Round to 2 decimal places.

A pharmacist wants to mix a \(30 \%\) saline solution with a \(10 \%\) saline solution to get \(200 \mathrm{~mL}\) of a \(12 \%\) saline solution. How much of each solution should she use?
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