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Chapter 4: Problem 12

Determine whether the ordered pair is a solution of the given system of equations. Remember to use alphabetical order of variables. $$ \begin{aligned} (-1,-2) ; & x+3 y=-7 \\ 3 x-2 y=& 12 \end{aligned} $$

Short Answer

Expert verified
No, the ordered pair (-1, -2) is not a solution to the system of equations.

Step by step solution

01

- Substitute the values into the first equation

Given the ordered pair (-1, -2) and the system of equations x + 3y = -7 and 3x - 2y = 12. Substitute x = -1 and y = -2 into the first equation: (-1) + 3(-2) = -7.
02

- Simplify the first equation

Simplify the expression in the first equation: -1 + 3(-2) = -1 - 6 = -7. Since -7 = -7, the ordered pair satisfies the first equation.
03

- Substitute the values into the second equation

Next, substitute x = -1 and y = -2 into the second equation: 3(-1) - 2(-2) = 12.
04

- Simplify the second equation

Simplify the expression in the second equation: 3(-1) - 2(-2) = -3 + 4 = 1. Since 1 eq 12 , the ordered pair does not satisfy the second equation.
05

Conclusion

Because the ordered pair (-1, -2) does not satisfy both equations in the system, it is not a solution.

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Key Concepts

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

ordered pairs
An ordered pair is a pair of numbers used to locate a point on a coordinate plane. It is written in the form \((x,y)\), where the first number represents the x-coordinate, and the second number is the y-coordinate. For example, in the ordered pair \((-1, -2)\), \(-1\) is the x-coordinate, and \(-2\) is the y-coordinate.
Ordered pairs can be used to check whether a specific point satisfies a system of equations by substituting the x and y values into each equation of the system. If all equations are satisfied, then the ordered pair is a solution to the system.
substitution method
The substitution method is a technique for solving systems of equations. To use this method, follow these steps:

1. **Solve one of the equations for one variable.**
2. **Substitute** this expression into the other equation to get an equation with only one variable.
3. **Solve the equation** for that variable.
4. **Substitute that solution** back into one of the original equations to find the value of the other variable.
5. **Check** the solution in both original equations to be sure it's correct.

For example, in our original exercise, we followed this method: We substituted \(-1\) for \ x\ and \(-2\) for \ y\ in the given equations to see if the ordered pair satisfies the system.
algebraic equations
An algebraic equation is a mathematical statement that shows the equality between two expressions. It often contains variables, numbers, and operations such as addition, subtraction, multiplication, and division. For instance, the equation \ x + 3y = -7 \ is an algebraic equation.

To determine if an ordered pair is a solution to a system of algebraic equations, you need to substitute the values into each equation:
  • Substitute \(-1\) for \ x \ and \(-2\) for \ y \ in \ x + 3y = -7 \ and simplify.
  • Check if the simplified result is equal to the right-side value of the equation.
  • Repeat the same steps for the second equation, \ 3x - 2y = 12 \.
In our exercise, after substitution, we found \ x + 3y = -7 \ is true but \ 3x - 2y e 12 \. So, the ordered pair \ (-1, -2) \ is not a solution to the system.

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Most popular questions from this chapter

Solve using a system of equations. The sum of two numbers is \(76 .\) One number is 2 more than the other. Find the numbers.

Solve each system using the substitution method. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this. $$ \begin{aligned} &x-3 y=7\\\ &-4 x+12 y=28 \end{aligned} $$

Solve each system using the substitution method. If a system has an infinite number of solutions, use set-builder notation to write the solution set. If a system has no solution, state this. $$ \begin{aligned} s &=\frac{1}{2} r \\ 3 r-4 s &=10 \end{aligned} $$

The Gathering Grounds charges \(\$ 10.00\) per pound for Columbian Supreme coffee and \(\$ 12.00\) per pound for Columbian Supreme decaffeinated coffee. The Lanosgas buy some of each to make their own partiallydecaffeinated blend. If they pay \(\$ 53\) for a 5 -lb blend, how much of each did they purchase?

Write at least three study tips of your own for someone beginning this exercise set.
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