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Chapter 5: Problem 4

Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations. $$ \left\\{\begin{array}{l} -3 x+y=8 \\ -x+2 y=-9 \end{array}\right. $$ (a) (-5,-7) (b) (-5,7)

Short Answer

Expert verified
(a) (-5, -7) is a solution. (b) (-5, 7) is not a solution.

Step by step solution

01

- Substitute the Ordered Pair

First, substitute the coordinates of the point into both equations of the system. Let's start with point (a) (-5,-7).For the first equation: -3x + y = 8 Substitute x = -5 and y = -7: -3(-5) + (-7) = 8 For the second equation: -x + 2y = -9 Substitute x = -5 and y = -7: -(-5) + 2(-7) = -9
02

- Solve the Substituted Equations for Point (a)

Now, solve each substituted equation to see if they hold true.First equation: -3(-5) + (-7) = 8 15 - 7 = 88 = 8 (True)Second equation: -(-5) + 2(-7) = -95 - 14 = -9-9 = -9 (True)Since both equations are true, the point (-5, -7) is a solution to the system.
03

- Substitute the Ordered Pair (b)

Next, substitute the coordinates of point (b) (-5,7) into both equations of the system.For the first equation: -3x + y = 8 Substitute x = -5 and y = 7: -3(-5) + 7 = 8For the second equation: -x + 2y = -9 Substitute x = -5 and y = 7: -(-5) + 2(7) = -9
04

- Solve the Substituted Equations for Point (b)

Now, solve each substituted equation to see if they hold true.First equation: -3(-5) + 7 = 815 + 7 = 822 鈮� 8 (False)Second equation: -(-5) + 2(7) = -95 + 14 = -919 鈮� -9 (False)Since both equations are false, the point (-5, 7) is not a solution to the system.

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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 set of two numbers, usually written in the form (x, y). The first number, x, is the coordinate on the horizontal axis, and the second number, y, is the coordinate on the vertical axis. These pairs are used to locate points on a graph.
In the context of systems of equations, ordered pairs are potential solutions. By substituting these values into the equations, you can check if they satisfy both equations.
For example, in the exercise above, we checked if the points (-5, -7) and (-5, 7) were solutions by substituting them into each equation.
This method helps to determine whether the coordinates of the ordered pair satisfy both equations simultaneously, proving if they are a valid solution or not.
substitution method
The substitution method is a technique for solving systems of equations. It involves substituting the value of one variable from one equation into the other equation. By doing so, you can reduce the system to a single equation with one variable, making it easier to solve.
  • First, solve one of the equations for one of the variables.
  • Then, substitute this value into the other equation.
  • Solve the resulting equation for the remaining variable.
  • Finally, substitute this value back into the original equation to find the other variable.
This method is useful for both linear and non-linear systems. By substituting the given ordered pairs into the equations in the example, we checked if the points satisfied the equations.
linear equations
A linear equation is an equation of the form ax + by = c, where a, b, and c are constants. In a graphical sense, a linear equation represents a straight line on a coordinate plane.
In systems of linear equations, we are usually dealing with two or more such equations. We look for points (ordered pairs) that lie on all lines represented by the equations. These points are the solutions to the system.
In the provided exercise, the system consists of two linear equations: -3x + y = 8 and -x + 2y = -9. We checked whether the ordered pairs (-5, -7) and (-5, 7) were solutions to these linear equations by substituting the x and y values into both equations. This demonstrates how linear equations can be used in systems to find common solutions.

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