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Chapter 7: Problem 39

Solve each system by any method. $$ \begin{array}{l} 2.2 x+1.3 y=-0.1 \\ 4.2 x+4.2 y=2.1 \end{array} $$

Short Answer

Expert verified
The solution is \( x = -\frac{5}{6}, y = \frac{4}{3} \).

Step by step solution

01

Multiply to Eliminate Decimals

First, we need to eliminate the decimals to make the system of equations easier to handle. Multiply the entire first equation by 10 to eliminate the decimals:\[ 10(2.2x + 1.3y) = 10(-0.1) \]\[ 22x + 13y = -1 \]For the second equation, multiply by 10 as well:\[ 10(4.2x + 4.2y) = 10(2.1) \]\[ 42x + 42y = 21 \] Now we have:\[\begin{align*}22x + 13y &= -1 \42x + 42y &= 21 \\end{align*}\]
02

Simplify the Second Equation

Notice that the second equation can be simplified by dividing all terms by 21:\[ \frac{42x}{21} + \frac{42y}{21} = \frac{21}{21} \]\[ 2x + 2y = 1 \]So the system becomes:\[\begin{align*}22x + 13y &= -1 \2x + 2y &= 1 \\end{align*}\]
03

Solve the System Using Substitution

To use substitution, solve the second equation for one variable. Let's solve for \(x\):\[ 2x + 2y = 1 \]\[ x = \frac{1 - 2y}{2} \]Substitute this expression for \(x\) into the first equation:\[ 22\left(\frac{1 - 2y}{2}\right) + 13y = -1 \]This simplifies to:\[ 11(1 - 2y) + 13y = -1 \]\[ 11 - 22y + 13y = -1 \]\[ 11 - 9y = -1 \]\[ -9y = -12 \]\[ y = \frac{4}{3} \]
04

Find the Value of x

Now that we have \( y = \frac{4}{3} \), substitute back into the expression for \( x \):\[ x = \frac{1 - 2\left(\frac{4}{3}\right)}{2} \]\[ x = \frac{1 - \frac{8}{3}}{2} \]\[ x = \frac{\frac{3}{3} - \frac{8}{3}}{2} \]\[ x = \frac{-\frac{5}{3}}{2} \]\[ x = -\frac{5}{6} \]
05

Verify the Solution

To confirm the solution \( x = -\frac{5}{6} \) and \( y = \frac{4}{3} \), substitute these values back into the original equations.For the first equation:\[ 2.2x + 1.3y = -0.1 \]\[ 2.2\left(-\frac{5}{6}\right) + 1.3\left(\frac{4}{3}\right) \approx -0.1 \]For the second equation:\[ 4.2x + 4.2y = 2.1 \]\[ 4.2\left(-\frac{5}{6}\right) + 4.2\left(\frac{4}{3}\right) = 2.1 \]Both equations are satisfied by these values, confirming the solution.

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Key Concepts

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

substitution method
When solving a system of equations, the substitution method is a powerful technique. It involves solving an equation for one variable and then using that expression to replace the variable in the other equation. This method allows you to simplify the system into a single equation that can be solved more easily.
Let's break it down further:
  • Solve for a Variable: Start by solving one of the equations for either variable. In our case, we simplified the second equation, \( 2x + 2y = 1 \), to express \( x \) in terms of \( y \). This gave us \( x = \frac{1 - 2y}{2} \).
  • Substitute: Take this expression and substitute it into the other equation. This allows us to eliminate one variable, as seen by plugging \( x \) into \( 22x + 13y = -1 \) which results in a single equation with just \( y \).
  • Solve for Remaining Variable: Once substituted, solve the resulting equation for the remaining variable, \( y \). This gave us \( y = \frac{4}{3} \).
By focusing on one variable at a time and substituting it into another equation, the substitution method simplifies a seemingly complex system into a manageable problem.
decimal elimination
When dealing with equations expressed in decimals, it can be challenging to solve them directly. Hence, one effective strategy is decimal elimination, which involves converting decimals into whole numbers for easier computation.
Here's how it works:
  • Multiply to Clear Decimals: Find a suitable factor, generally by multiplying by 10 or 100, to turn all values into whole numbers. In the given equations, multiplying each term by 10 helped us to transform \( 2.2x + 1.3y = -0.1 \) into \( 22x + 13y = -1 \), and \( 4.2x + 4.2y = 2.1 \) into \( 42x + 42y = 21 \).
  • Simplify Further: Once decimals are cleared, additional simplifications may be possible. For example, dividing the whole number equation \( 42x + 42y = 21 \) by 21 gave us \( 2x + 2y = 1 \), which is simpler to work with.
Removing decimals not only makes arithmetic calculations easier but also reduces the potential for arithmetic errors, making the entire solving process much smoother.
verification of solutions
After finding potential solutions to a system of equations, it's crucial to verify that they are correct. This step ensures the solutions satisfy the original conditions set by the equations, confirming their validity.
Let's see the verification process:
  • Substitute Back: Take the calculated values, in this case, \( x = -\frac{5}{6} \) and \( y = \frac{4}{3} \), and substitute them back into the original equations. You should check both equations to ensure complete accuracy.
  • Check Each Equation: Verify whether these values satisfy both original equations. For the first equation, when substituting \( 2.2(-\frac{5}{6}) + 1.3(\frac{4}{3}) \), we calculate approximately \(-0.1\), which is in line with the original equation's requirement. Similarly, confirming \( 4.2(-\frac{5}{6}) + 4.2(\frac{4}{3}) = 2.1 \) satisfies the second equation.
By substituting the variables' values back into the initial equations and checking for consistency, you reaffirm that the solutions are indeed correct. Verification is an essential step to ensure the accuracy and reliability of your solutions in problem-solving.

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Most popular questions from this chapter

For the following exercises, use Gaussian elimination to solve the system. $$ \begin{aligned} \frac{x-3}{4}-\frac{y-1}{3}+2 z &=-1 \\ \frac{x+5}{2}+\frac{y+5}{2}+\frac{z+5}{2} &=7 \\ x+y+z &=1 \end{aligned} $$

For the following exercises, solve the system by Gaussian elimination. $$ \left[\begin{array}{rrr|r} -0.1 & 0.3 & -0.1 & 0.2 \\ -0.4 & 0.2 & 0.1 & 0.8 \\ 0.6 & 0.1 & 0.7 & -0.8 \end{array}\right] $$

For the following exercises, create a system of linear equations to describe the behavior. Then, solve the system for all solutions using Cramer鈥檚 Rule. You invest \(\$ 10,000\) into two accounts, which receive 8\(\%\) interest and 5\(\%\) interest. At the end of a year, you had \(\$ 10,710\) in your combined accounts. How much was invested in each account?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. You invested \(\$ 10,000\) into two accounts: one that has simple \(3 \%\) interest, the other with \(2.5 \%\) interest. If your total interest payment after one year was \(\$ 283.50\), how much was in each account after the year passed?

For the following exercises, set up the augmented matrix that describes the situation, and solve for the desired solution. A bag of mixed nuts contains cashews, pistachios, and almonds. There are \(1,000\) total nuts in the bag, and there are 100 less almonds than pistachios. The cashews weigh 3 \(\mathrm{g}\) , pistachios weigh 4 \(\mathrm{g}\) , and almonds weigh 5 \(\mathrm{g}\) . If the bag weighs \(3.7 \mathrm{kg},\) find out how many of each type of nut is in the bag.
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