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Chapter 6: Problem 32

Solve each system by elimination. $$\begin{array}{r}5 x+7 y=6 \\\10 x-3 y=46\end{array}$$

Short Answer

Expert verified
The solution is \(x = 4\) and \(y = -2\).

Step by step solution

01

Align Equations

Write the system of equations in a column format for clarity:1. \(5x + 7y = 6\)2. \(10x - 3y = 46\).
02

Equalize Coefficients of x

Multiply the first equation by 2 so that the coefficients of \(x\) in both equations match:\[2(5x + 7y) = 2(6)\]This gives us the new equation:\(10x + 14y = 12\).
03

Subtract Equations to Eliminate x

Subtract the second equation from this new equation to eliminate \(x\):\[(10x + 14y) - (10x - 3y) = 12 - 46 \]Resulting in:\(17y = -34\).
04

Solve for y

Divide both sides of the equation \(17y = -34\) by 17 to solve for \(y\):\[y = \frac{-34}{17} = -2\].
05

Substitute y back into one of the original equations

Use \(y = -2\) in the first original equation \(5x + 7y = 6\):\[5x + 7(-2) = 6\]Simplify to find \(x\):\[5x - 14 = 6\].
06

Solve for x

Add 14 to both sides to isolate \(5x\):\[5x = 20\]Then divide by 5:\[x = \frac{20}{5} = 4\].
07

Solution

The solution to the system of equations is \(x = 4\) and \(y = -2\).

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Key Concepts

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

Elimination Method
The elimination method is a popular technique used to solve systems of equations. Its primary goal is to eliminate one variable so that you can easily solve for the other. Using this method involves combining equations strategically. You can do this by either addition or subtraction once the coefficients of the variables are aligned.

Here's a quick recap:
  • Align Equations: Start by writing the system in a column format. This helps you visually track the coefficients and corresponding terms.
  • Matching Coefficients: Adjust the coefficients of one variable so they match, letting you eliminate it by subtraction or addition.
  • Eliminate and Solve: Once a variable is eliminated, solve the resulting simple equation for the remaining variable.
By following these steps, you'll find the values for both variables efficiently. It's like peeling the layers of an onion until you're left with the core鈥攖he solution!
Algebra
Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It's foundational in high school mathematics and involves working with equations, like our system of equations exercise.

In a system of equations:
  • Expressions: Algebra involves expressions that can include constants, variables, and operations (like plus or minus).
  • Equations: Equations are statements of equality with expressions on both sides. Solving them involves finding the values for variables that satisfy this equality.
  • Manipulations: Algebra teaches us how to rearrange and simplify these expressions, making the unknown variables easier to solve for.
Think of algebra as a toolbox that equips you with the techniques needed to crack any math puzzle. By mastering algebra, you transform complex problems into manageable solutions.
Solving Equations
Solving equations is a vital skill, and one of the core concepts in algebra. It allows you to find the unknown values that make an equation true. Here's a breakdown of the process as demonstrated in our exercise:

  • Transforming Equations: Begin by transforming equations by multiplying or adding to align terms. This makes them easier to handle.
  • Isolating Variables: Use strategic operations to isolate the variable. This might involve adding, subtracting, multiplying, or dividing both sides by the same number.
  • Substitution: Once you solve for one variable, substitute it into another equation. This leads you to solve for the remaining variable.
These steps form a solid framework for tackling various math problems. With practice, solving equations becomes an intuitive and streamlined process, paving the way for understanding more complex mathematical concepts.

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Most popular questions from this chapter

Solve each application. Financing Expansion To get funds necessary for a planned expansion, a small company took out three loans totaling \(\$ 12,500 .\) The company was able to borrow some of the money at \(2 \% .\) It borrowed \(\$ 1000\) more than \(\frac{1}{2}\) the amount of the \(2 \%\) loan at \(3 \%\) and the rest at \(2.5 \% .\) The total annual interest was \(\$ 305 .\) How much did the company borrow at each rate?

To analyze population dynamics of the northern spotted owl, mathematical ecologists divided the female owl population into three categories: juvenile (up to 1 year old), subadult (1 to 2 years old), and adult (over 2 years old). They concluded that the change in the makeup of the northern spotted owl population in successive years could be described by the following matrix equation. $$\left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=\left[\begin{array}{rrr} 0 & 0 & 0.33 \\ 0.18 & 0 & 0 \\ 0 & 0.71 & 0.94 \end{array}\right]\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right]$$ The numbers in the column matrices give the numbers of females in the three age groups after \(n\) years and \(n+1\) years. Multiplying the matrices yields the following. $$\begin{aligned} &j_{n+1}=0.33 a_{n}\\\ &s_{n+1}=0.18 j_{n}\\\ &a_{n+1}=0.71 s_{n}+0.94 a_{n} \end{aligned}$$ (Source: Lamberson, R. H., R. McKelvey, B. R. Noon, and C. Voss, "A Dynamic Analysis of Northern Spotted Owl Viability in a Fragmented Forest Landscape," Conservation Biology, Vol. \(6, \text { No. } 4 .)\) (a) Suppose there are currently 3000 female northern spotted owls: 690 juveniles, 210 subadults, and 2100 adults. Use the preceding matrix equation to determine the total number of female owls for each of the next 5 years. (b) Using advanced techniques from linear algebra, we can show that, in the long run, $$ \left[\begin{array}{c} j_{n+1} \\ s_{n+1} \\ a_{n+1} \end{array}\right]=0.98359\left[\begin{array}{c} j_{n} \\ s_{n} \\ a_{n} \end{array}\right] $$ What can we conclude about the long-term fate of the northern spotted owl? (c) In this model, the main impediment to the survival of the northern spotted owl is the number 0.18 in the second row of the \(3 \times 3\) matrix. This number is low for two reasons: The first year of life is precarious for most animals living in the wild, and juvenile owls must eventually leave the nest and establish their own territory. If much of the forest near their original home has been cleared, then they are vulnerable to predators while searching for a new home. Suppose that, due to better forest management, the number 0.18 can be increased to \(0.3 .\) Rework part (a) under this new assumption.

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$y \leq x^{3}+x^{2}-4 x-4$$

Solve each system. Write solutions in terms of \(z\) if necessary. $$\begin{aligned} 3 x+6 y-3 z &=12 \\ -x-2 y+z &=16 \\ x+y-2 z &=20 \end{aligned}$$

The average of self-reported spending "yesterday" for high-income consumers and middle-/low-income consumers was 93.50 dollars in September 2012 . High- income consumers spend 65 dollars more than middle-/low-income consumers. (Source: www.marketingcharts.com) (a) Write a system of equations whose solution gives the self-reported spending for each income group. Let \(x\) be the spending by high-income consumers and \(y\) be the spending by middle-/low-income consumers. (b) Solve the system. (c) Interpret the solution.
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