Problem 60 - Algebra 2

Question

Solve each system by substitution. $$ \left\\{\begin{array}{l}{x=5} \\ {x-y+z=5} \\ {x+y-z=-5}\end{array}\right. $$

Step-by-Step Solution

Answer

The system has no solution because when we substitute x with 5 and simplify the equations, adding them together leads to a contradiction, 0 = -10.

Step 1: Identify the equation to substitute

Since the first equation gives us the value of x directly, there is no need for manipulation. We can use the value of x directly in substitution for the other two equations.

Step 2: Substitute the value of x into the second and third equations

Replace x with 5 in the second and third equations to simplify them into two equations with two variables, y and z.

Step 3: Simplify the two equations after substitution

After substituting x with 5, the second equation becomes 5 - y + z = 5, which simplifies to -y + z = 0. The third equation becomes 5 + y - z = -5, which simplifies to y - z = -10.

Step 4: Solve the simplified system for y and z

Now we have a system of two equations with two variables: 1) -y + z = 0, 2) y - z = -10. Add both equations to eliminate z, resulting in -y + y + z - z = 0 - 10.

Step 5: Solve for y

After adding the equations, we get 0 = -10, which is not possible with the given system. This means the system has no solution.

Key Concepts

Systems of Linear Equations

A system of linear equations consists of two or more linear equations that are related by having one or several solutions, where their graphs intersect.

A linear equation in two or three variables corresponds to a geometric line or plane, respectively. A solution to a system is a set of values for the variables that satisfy all equations simultaneously. When working with systems of equations, the goal is to find the coordinate(s) at which all the equations intersect.

In practice, there are several types of outcomes for a system of linear equations: one unique solution, infinitely many solutions, or no solution. One unique solution occurs when the lines or planes intersect at a single point. Infinitely many solutions happen when the equations describe the same line or plane. On the other hand, if the lines or planes do not intersect at all, which means they are parallel, it results in a system with no solution.

Substitution Method

The substitution method is a technique for solving a system of linear equations. This method involves solving one of the equations for one variable and then substituting the resulting expression into the other equation(s).

Here’s how it generally works in a step-wise approach:

• Choose an equation with a variable that can be conveniently isolated.
• Solve this equation for that particular variable to obtain an expression.
• Substitute this expression into the other equation(s) in place of the variable.
• Solve the resulting equation for the other variable(s).
• Back-substitute the found values into the initial equations wherever necessary to find all variable values.

The method is particularly useful when one equation in the system can be easily rearranged to isolate one variable. This strategy simplifies the system to one with fewer equations and variables, making it easier to solve.

No Solution System

A system of equations that has no solution is one where the equations represent lines or planes that never intersect; they are parallel to each other. This concept is also known as an inconsistent system.

When using the substitution method, you might arrive at an equation that simplifies to a false statement, such as in our exercise - we simplified the system to get a final equation of \(0 = -10\). This contradiction indicates that no values exist for the variables that can make both (or all) original equations true simultaneously.

In cases where you obtain an absurd or impossible result, it's important not to assume a mistake in your calculations too hastily. Although it's always a good idea to double-check your work, such a result may be correct and reveals the nature of the system itself - that it genuinely has no solution.