Problem 162 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 5 x-8 y=12 \\ 10 x-16 y=20 \end{array}\right. $$

Step-by-Step Solution

Answer

The system has no solution.

Step 1: Simplify the System

Observe the given system of equations:\[\begin{cases} 5x - 8y = 12 \ 10x -16y = 20 \end{cases}\]Notice that the second equation is a multiple of the first equation. Divide the second equation by 2 for simplification:\[ 5x - 8y = 10 \]

Step 2: Combine the Equations

Rewriting both equations, we now have:\[\begin{cases} 5x - 8y = 12 \ 5x - 8y = 10 \end{cases}\]Since the left-hand sides of both equations are identical but the right-hand sides are different, the system has no solution.

Key Concepts

systems of linear equations

A system of linear equations consists of two or more linear equations that share common variables.
Solving such a system means finding the values of the variables that satisfy all the equations simultaneously.
For example, in the given exercise, we have the system: \[ \begin{cases} 5x - 8y = 12 \ 10x - 16y = 20 \end{cases} \]
This means we need to find values for x and y that make both equations true at the same time.
Linear equations can be presented in various forms, such as:

• Slope-intercept form: \text{y = mx + b}\
• Standard form: \text{Ax + By = C}\

In this exercise, both equations are in the standard form.
The solution can either be a single point where both lines intersect, infinitely many points if the lines are the same, or no points if the lines are parallel but not identical.

substitution method

The substitution method is one way to solve a system of linear equations.
It involves solving one of the equations for one variable and then substituting that expression into the other equation.
Let's look at the given system again:\[ \begin{cases} 5x - 8y = 12 \ 10x - 16y = 20 \end{cases} \]
Here’s how we apply substitution method:

• First, we notice that the second equation is just a multiple of the first one. This suggests a special case may occur.
• For simplicity, let’s divide the second equation by 2:
\[ 5x - 8y = 10 \]
This simplifies our system to:\[ \begin{cases} 5x - 8y = 12 \ 5x - 8y = 10 \end{cases} \]
Notice we have two identical terms on the left but different constants on the right.
This contradiction shows that there is no value of x and y that can satisfy both equations simultaneously.
Thus, we conclude that this system has no solution.

inconsistent systems

An inconsistent system is a system of linear equations with no solution.
This occurs when the equations represent parallel lines that never intersect.
In our example, after simplifying, we found:\[ \begin{cases} 5x - 8y = 12 \ 5x - 8y = 10 \end{cases} \]
Both equations have the same left-hand side but different right-hand sides, indicating parallel lines.
Since the lines are parallel and not the same, they will never meet, meaning the system has no solution.
Here are a few key points about inconsistent systems:

• They occur when the equations describe parallel lines.
• There is no point (x, y) that will satisfy both equations.
• Graphing the lines will show that they never intersect.

Understanding inconsistent systems helps in determining when a system of equations doesn't have any solution.