Problem 58
Question
Use the substitution method or linear combinations to solve the linear system and tell how many solutions the system has. $$\begin{aligned} -2 x+2 y &=4 \\ x-y &=-2 \end{aligned}$$
Step-by-Step Solution
Verified Answer
The system of equations has an infinite number of solutions.
1Step 1: Simplify The Equations
The given equations can be simplified for easier computation. Divide the first equation by 2 to get: -x + y = 2. So, the system of equations becomes: \n (1) -x + y = 2 and (2) x - y = -2.
2Step 2: Use Addition Method
The addition (or linear combination) method involves adding the two equations to eliminate one of the variables. Here, if we add the two simplified equations: \n (-x + y) + (x - y) = (2 - 2)\n The x's and y's cancel out, resulting in 0 = 0.
3Step 3: Interpret the Result
The statement 0 = 0 is always true. This means the original system of equations has an infinite number of solutions because there are countless pairs of (x, y) that will satisfy both equations. This typically indicates that the equations represent the same line in a graph.
Key Concepts
Substitution MethodLinear Combination MethodSystems of EquationsInfinite Solutions
Substitution Method
The substitution method is a technique used to solve systems of equations by finding the value of one variable and then substituting it into the other equation. To understand this method, consider a system of two equations with two variables.
The goal is to isolate one variable in one equation and then replace its value in the second equation. For example, if one equation is solved for variable x, that expression for x is used in place of the x in the other equation. This eventually leads to an equation with one variable, which can be solved easily.
When applying this method to our problem, we could solve the second equation for x, obtaining x = y - 2, and then substitute this expression into the first equation. However, this method is not suitable for all systems, such as those with equations that are multiples of each other or those that result in an identity or a contradiction when using substitution.
The goal is to isolate one variable in one equation and then replace its value in the second equation. For example, if one equation is solved for variable x, that expression for x is used in place of the x in the other equation. This eventually leads to an equation with one variable, which can be solved easily.
When applying this method to our problem, we could solve the second equation for x, obtaining x = y - 2, and then substitute this expression into the first equation. However, this method is not suitable for all systems, such as those with equations that are multiples of each other or those that result in an identity or a contradiction when using substitution.
Linear Combination Method
Also known as the addition method, the linear combination method is utilized when we work with a system of equations to eliminate one of the variables. By carefully choosing coefficients, we can add or subtract the equations to cancel out one of the variables.
In our exercise, we added the simplified equations (-x + y) and (x - y) directly, since the coefficients of x were already set to cancel each other out. Together with the y variables, this yielded 0 = 0. This unique outcome indicates that the system has either no solution or infinitely many solutions, which we explore further in the context of our exercise. This method is particularly efficient when equations are arranged in a way that makes one of the variables easy to eliminate.
In our exercise, we added the simplified equations (-x + y) and (x - y) directly, since the coefficients of x were already set to cancel each other out. Together with the y variables, this yielded 0 = 0. This unique outcome indicates that the system has either no solution or infinitely many solutions, which we explore further in the context of our exercise. This method is particularly efficient when equations are arranged in a way that makes one of the variables easy to eliminate.
Systems of Equations
A system of equations is a set of two or more equations with the same variables that are to be solved simultaneously. These systems can have one solution, no solution, or infinitely many solutions.
When solving for systems, the solution(s) are the point(s) where the graphs of the equations intersect. If the system has one solution, it is consistent and the equations graph as intersecting lines. If there are no solutions, the system is inconsistent; the equations represent parallel lines that never meet. Lastly, if there are infinitely many solutions, the system is dependent and the equations are multiple representations of the same line.
When solving for systems, the solution(s) are the point(s) where the graphs of the equations intersect. If the system has one solution, it is consistent and the equations graph as intersecting lines. If there are no solutions, the system is inconsistent; the equations represent parallel lines that never meet. Lastly, if there are infinitely many solutions, the system is dependent and the equations are multiple representations of the same line.
Infinite Solutions
Infinite solutions occur in a system of equations when the two equations are not distinct but rather multiples of each other, representing the same line in a graph. This implies that every point on the line is a solution to both equations.
In our given exercise, the simplification of the equations and the subsequent addition resulted in 0 = 0, an identity that holds true for all real numbers. Therefore, we conclude that there are an infinite number of solutions to the system. To visualize this, one might graph both equations and see that they are, in fact, the same line, which is why each point on that line satisfies the original system of equations.
In our given exercise, the simplification of the equations and the subsequent addition resulted in 0 = 0, an identity that holds true for all real numbers. Therefore, we conclude that there are an infinite number of solutions to the system. To visualize this, one might graph both equations and see that they are, in fact, the same line, which is why each point on that line satisfies the original system of equations.
Other exercises in this chapter
Problem 58
Write the percent as a fraction or as a mixed number in simplest form. (Skills Review p. 768 ) $$ 6 \% $$
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Find the x-intercepts of the graph of the function. $$y=-x^{2}+x+6$$
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Graph the inequality. \(-2 x-y \geq 4\)
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