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Chapter 9: Problem 27

Solve each system by elimination. First clear denominators. $$\begin{array}{l} \frac{x}{2}+\frac{y}{3}=4 \\ \frac{3 x}{2}+\frac{3 y}{2}=15 \end{array}$$

Short Answer

Expert verified
(x, y) = (4, 6)

Step by step solution

01

- Clear the Denominators

Multiply each equation by a common multiple of the denominators to clear fractions. Multiply the first equation by 6, and the second equation by 2.
02

- Simplify the Equations

After multiplication, the system of equations becomes: \[3x + 2y = 24\] \[3x + 3y = 30\]
03

- Eliminate One Variable

Subtract the first equation from the second equation to eliminate the variable \(x\). \[(3x + 3y) - (3x + 2y) = 30 - 24\] \[y = 6\]
04

- Solve for the Other Variable

Substitute \(y = 6\) into the first equation: \[3x + 2(6) = 24 \] \[3x + 12 = 24 \] \[3x = 12 \] \[x = 4\]

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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 for solving systems of equations. It involves eliminating one of the variables by adding or subtracting the equations. This simplifies the system and makes it easier to solve for the remaining variable.

Let's break it down further:
  • First, align both equations vertically, ensuring corresponding variables and constants are in the same column.
  • Then, multiply one or both equations by suitable numbers so the coefficients of one of the variables are equal (but opposite in sign, if needed).
  • Next, add or subtract the equations to eliminate that variable.
  • Finally, solve the remaining equation for the other variable, and substitute back to find the eliminated variable.
Elimination is particularly useful when the system's equations are already lined up and when the coefficients easily lend themselves to cancellation.
clearing denominators
Clearing denominators is a crucial first step in solving systems of equations that involve fractions. This process makes the subsequent calculations easier and reduces errors.

Here's how to clear denominators effectively:
  • Identify the least common multiple (LCM) of the denominators in the system.
  • Multiply every term of each equation by the LCM. This step eliminates the fractions, converting them to whole numbers.
  • This results in a simpler system with integer coefficients, making it easier to use methods like elimination or substitution.
In the example provided, we cleared the denominators by multiplying the first equation by 6 and the second equation by 2, giving us whole-number coefficients and making the next steps straightforward.
substitution method
The substitution method is another popular technique for solving systems of equations. It involves solving one equation for one variable and then substituting that solution into the other equation.

Let's see how to use this method:
  • Solve one of the equations for one of the variables in terms of the other variable.
  • Substitute this expression into the other equation. This will give you an equation with a single variable.
  • Solve this single-variable equation to find the value of one variable.
  • Substitute this value back into one of the original equations to solve for the other variable.
Substitution is especially useful when one of the equations is easily solvable for one variable, making it straightforward to substitute and solve.

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Most popular questions from this chapter

For each pair of matrices \(A\) and \(B,\) find \((a) A B\) and \((b) B A\). $$A=\left[\begin{array}{rrr} -1 & 0 & 1 \\ 0 & 1 & 1 \\ -1 & -1 & 0 \end{array}\right], B=\left[\begin{array}{lll} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\right]$$

For certain aircraft there exists a quadratic relationship between an airplane's maximum speed \(S\) (in knots) and its ceiling \(C\), or highest altitude possible (in thousands of feet). The table lists three airplanes that conform to this relationship. $$\begin{array}{|c|c|c} \hline \text { Airplane } & \text { Max Speed (S) } & \text { Ceiling (C) } \\\ \hline \text { Hawkeye } & 320 & 33 \\ \hline \text { Corsair } & 600 & 40 \\ \hline \text { Tomcat } & 1283 & 50 \\ \hline \end{array}$$ (a) If the quadratic relationship between \(C\) and \(S\) is written as \(C=a S^{2}+b S+c\) use a system of linear equations to determine the constants \(a, b\) and \(c,\) and give the equation. (b) A new aircraft of this type has a ceiling of \(45,000 \mathrm{ft}\). Predict its top speed.

Solve each problem. In certain parts of the Rocky Mountains, deer provide the main food source for mountain lions. When the deer population is large, the mountain lions thrive. However, a large mountain lion population reduces the size of the deer population. Suppose the fluctuations of the two populations from year to year can be modeled with the matrix equation $$ \left[\begin{array}{c} m_{n+1} \\ d_{n+1} \end{array}\right]=\left[\begin{array}{rr} 0.51 & 0.4 \\ -0.05 & 1.05 \end{array}\right]\left[\begin{array}{l} m_{n} \\ d_{n} \end{array}\right] $$ The numbers in the column matrices give the numbers of animals in the two populations after \(n\) years and \(n+1\) years, where the number of deer is measured in hundreds. (a) Give the equation for \(d_{n+1}\) obtained from the second row of the square matrix. Use this equation to determine the rate at which the deer population will grow from year to year if there are no mountain lions. (b) Suppose we start with a mountain lion population of 2000 and a deer population of \(500,000\) (that is, 5000 hundred deer). How large would each population be after 1 yr? 2 yr? (c) Consider part (b) but change the initial mountain lion population to \(4000 .\) Show that the populations would both grow at a steady annual rate of 1.01

Solve each problem. The Fan Cost Index (FCI) is a measure of how much it will cost a fam- ily of four to attend a professional sports event. In \(2010,\) the FCI prices for Major League Baseball and the National Football League averaged \(\$ 307.76 .\) The FCI for baseball was \(\$ 225.56\) less than that for football. What were the FCIs for these sports? (Source: Team Marketing Report.)

Solve each system by using the inverse of the coefficient matrix. $$\begin{aligned} &x+3 y=-12\\\ &2 x-y=11 \end{aligned}$$
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