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Chapter 4: Problem 26

(a) solve by elimination. (b) if there is one solution, check. $$ \begin{aligned} &3 x+2 y=-6 \\ &4 x+5 y=-1 \end{aligned} $$

Short Answer

Expert verified
The solution is (x, y) = (-4, 3). Checking the solution verifies its correctness.

Step by step solution

01

- Multiply to Eliminate One Variable

To eliminate one of the variables, make the coefficients of either x or y the same. Multiply the first equation by 4 and the second equation by 3.
02

- Apply Multiplication

Multiply equation 1 by 4: \[ 4(3x + 2y) = 4(-6) \] \[ 12x + 8y = -24 \] Multiply equation 2 by 3: \[ 3(4x + 5y) = 3(-1) \] \[ 12x + 15y = -3 \]
03

- Subtract the Equations

Subtract the second new equation from the first new equation:\[ (12x + 8y) - (12x + 15y) = -24 - (-3) \] \[ -7y = -21 \]
04

- Solve for y

Divide both sides of the equation by -7:\[ y = 3 \]
05

- Substitute y into One of the Original Equations

Substitute y = 3 into the first original equation: \[ 3x + 2(3) = -6 \] \[ 3x + 6 = -6 \]
06

- Solve for x

Subtract 6 from both sides of the equation: \[ 3x = -12 \] Divide by 3: \[ x = -4 \]
07

- Check the Solution

Substitute x = -4 and y = 3 into the second original equation to verify: \[ 4(-4) + 5(3) = -1 \] \[ -16 + 15 = -1 \] The solution is verified.

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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 used to solve systems of linear equations. It involves manipulating the equations to cancel out one variable, making it easier to solve for the other.

The steps in the elimination method usually include:
  • Making the coefficients of one of the variables (either x or y) the same in both equations.
  • Adding or subtracting the equations to eliminate that variable.
  • Solving the resulting single-variable equation.

In the given exercise, we first multiply the equations so that the coefficients of x become equal. This allows us to eliminate x by subtracting the two equations.
Solving Linear Equations
Once you've eliminated one of the variables using the elimination method, you'll be left with a single linear equation.
Linear equations are equations of the first order, meaning they have no exponents higher than one.
To solve these, the primary steps are:
  • Isolate the variable by using basic algebraic operations such as addition, subtraction, multiplication, and division.
  • Once the variable is isolated on one side of the equation, you can find its value easily.

In our example, after eliminating x, we get a simplified equation in terms of y, which we solve to find y.
Substitution Method
The substitution method is another approach to solving systems of equations.
Here, one equation is solved for one variable in terms of the other variable, and then this expression is substituted into the other equation. Although not primarily used in this exercise, substitution can complement elimination:
  • First, solve one of the equations for one of the variables.
  • Next, substitute this expression into the other equation.
  • Finally, solve the resulting equation for the remaining variable.

This method is useful when one equation is easy to solve for one of the variables.
Checking Solutions
Checking the solution of a system of equations ensures that our answers are correct. To check:
  • Substitute the found values of x and y back into both original equations.
  • Verify that both equations are satisfied with these values.

For our solution, we found x = -4 and y = 3. Substituting these values back into both original equations confirms they are correct, as both sides of the equations equal after the substitution.

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Most popular questions from this chapter

Snack Mix A is \(17 \%\) raisins. Snack Mix B is \(8 \%\) raisins. Find the amount of Snack Mix A and Snack Mix B needed to make 30 pounds of a new snack mix that is \(12 \%\) raisins. Round to the nearest whole number.

For exercises 55-58, (a) Write four inequalities that represent the constraints. (b) Graph the inequalities that represent the constraints. Label the feasible region. An investor will put a maximum of $$\$ 25,500$$ in foreign investments and domestic investments with at least four times as much in domestic investments as in foreign investments and a minimum of $$\$ 2000$$ in foreign investments. Let \(x=\) amount in foreign investments, and let \(y=\) amount of domestic investments.

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