Problem 24 - Introductory Algebra for College Students

Question

Solve each system by the substitution method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x=5 y-3 \\\x=8 y+4\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(\{(-11, -7/3)\}\).

Step 1: Identify an Equation to Substitute

First observe the system of equations which are \(x = 5y - 3\) and \(x = 8y + 4\). We can see that both equations are already solved for \(x\). So we can directly substitute \(x\)from the first equation into the second equation. That is, wherever we see \(x\) in the second equation, we can replace it with \(5y - 3\).

Step 2: Solve for the Variable

Now in the second equation, we replace \(x\) with \(5y - 3\) to get \(5y - 3 = 8y + 4\). Let's simplify this to solve for \(y\). Start by subtracting \(5y\) from both sides to get \(-3 = 3y + 4\). Then subtract 4 from both sides to get \(-7 = 3y\). Dividing both sides of the equation \(-7 = 3y\) by 3 gives us \(y = -7/3\).

Step 3: Find the value of the first variable

Having solve for \(y\), we can find the value of \(x\) by substituting \(-7/3\) into either of the original equations. Let's use the first equation \(x = 5y - 3\). So substituting \(y = -7/3\) in \(x = 5y - 3\) gives us \(x = 5(-7/3) - 3 = -11\).

Step 4: Present the solution

Now we have \(x = -11\) and \(y = -7/3\). So the solution to the system can be represented in set notation as \(\{(-11, -7/3)\}\)

Key Concepts

Substitution Method

The substitution method is a technique used to solve systems of linear equations. It involves replacing one variable with a corresponding expression to find the value of the other variable. This method can be very efficient when one of the equations is already solved for a variable. In our example, both equations are solved for the variable \(x\), making it easy to substitute.To use the substitution method:

• Identify one equation solved for a variable. In our problem, both equations were given in terms of \(x\): \(x = 5y - 3\) and \(x = 8y + 4\).
• Substitute this expression into the other equation. This means wherever \(x\) appears in the second equation, replace it with \(5y - 3\).
• Solve the resulting equation for the other variable \(y\).
• Finally, substitute back to find the remaining variable \(x\).

By eliminating variables step by step, the substitution method reduces complexity and makes solving linear systems manageable.

Set Notation

Set notation is a standardized way to express solutions to equations, especially when dealing with systems of equations. It is a concise way to show the values that satisfy all equations in the system. In the example solution, set notation was used to present the solution as \(\{(-11, -7/3)\}\).When you use set notation:

• Wrap the elements of your solution, often as ordered pairs, in curly braces \(\{ \} \).
• The order of the elements is crucial, especially in systems like pairs \((x, y)\).
• This notation communicates that the solution fits all given equations.

Set notation is particularly effective for conveying the solution succinctly, whether there's one solution, no solution, or infinitely many solutions.

Linear Equations

Linear equations form the foundation of solving systems of equations. They are equations where the variables are raised to no powers higher than one. Linear equations can be graphed as straight lines, and their solutions can often be found at the intersections of these lines.In our problem, we dealt with two linear equations, both in the form:

• \(x = 5y - 3\)
• \(x = 8y + 4\)

This form is highly cooperative with the substitution method because one variable, \(x\), is already isolated. Linear equations can have unique solutions, no solution, or infinitely many solutions.

• If the lines intersect at one point, there is one unique solution.
• If the lines are parallel (no intersection), there is no solution.
• If the lines coincide (overlap perfectly), there are infinitely many solutions.

Understanding the nature of linear equations is key to predicting the type of solution a system may have.