Problem 165 - Elementary Algebra

Question

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 5 x-3 y=15 \\ y=\frac{5}{3} x-2 \end{array}\right. $$

Step-by-Step Solution

Answer

No solutions. The equations are inconsistent.

Step 1: Identify the equations

The system of equations is: 1. \( 5x - 3y = 15 \) 2. \( y = \frac{5}{3}x - 2 \)

Step 2: Substitute the second equation into the first equation

Replace \( y \) in the first equation with \( \frac{5}{3}x - 2 \):\[ 5x - 3(\frac{5}{3}x - 2) = 15 \]

Step 3: Simplify the equation

Distribute and simplify the equation: \[ 5x - 3 \times \frac{5}{3}x + 6 = 15 \] Simplify further: \[ 5x - 5x + 6 = 15 \] \[ 6 = 15 \]

Step 4: Analyze the result

Since the simplified equation results in a contradiction (6 does not equal 15), there are no solutions to this system of equations.

Key Concepts

Substitution Method

The basic idea behind the substitution method is to solve one of the equations in a system for one variable. You then substitute this expression into the other equation. This allows you to solve for the remaining variable. In our example:

• The first equation is: \(5x - 3y = 15\)
• The second equation is: \(y = \frac{5}{3}x - 2\)

To use substitution, we note that the second equation expresses \(y\) in terms of \(x\). So, we substitute \(\frac{5}{3}x - 2\) for \(y\) in the first equation.
By following these steps, you can simplify the system to a single-variable equation, which makes it easier to solve.

Algebraic Equations

Algebraic equations involve variables and constants combined using mathematical operations. In solving systems of algebraic equations, such as our given system:

• First equation: \(5x - 3y = 15\)
• Second equation: \(y = \frac{5}{3}x - 2\)

We need to perform algebraic manipulations to find the variable values. These manipulations include distributing, combining like terms, and isolating variables.
In the substitution step, we replace \(y\) in the first equation with \(\frac{5}{3}x - 2\), then distribute and combine like terms to simplify the equation:
\[\begin{align*} 5x - 3\left( \frac{5}{3}x - 2 \right) &= 15ewline5x - 5x + 6 &= 15ewline6 &eq 15\end{align*}\] Since we end up with a contradiction, it indicates an inconsistency in the equations.

No Solution Systems

A 'no solution' system means that there's no set of values for the variables that satisfies all the given equations. When you end up with a false statement like \(6 = 15\), it tells you that the lines representing the equations never intersect. This happens in parallel lines, which never meet.
For our system:

• First equation's line: \(5x - 3y = 15\)
• Second equation's line: \(y = \frac{5}{3}x - 2\)

When substituting \(\frac{5}{3}x - 2\) into the first equation, we saw that the left-side simplified to a false statement when we tried to isolate \(x\). This contradiction confirms they are parallel since they have the same slope (both get reduced to \(\frac{5}{3}\)) but different y-intercepts.
Recognizing these consistent patterns helps in identifying and solving different types of linear system problems.