Problem 30 - Introductory Algebra for College Students

Question

Solve each system by the substinuion method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets.$ $$\left\\{\begin{array}{l}\frac{x}{4}-\frac{y}{4}=-1 \\\x+4 y=-9\end{array}\right.$$

Step-by-Step Solution

Answer

The solution set for the given system of equations is {(x, y)| x = -5 and y = -1}

Step 1: Express one variable in terms of the other from Equations

Let's start by expressing x in terms of y from the second equation. By rearranging the second equation \(x+4y=-9\) we can get \(x=-4y -9\)

Step 2: Substitute the above expression in the first equation

Substitute the expression for x from step 1 into the first equation. The equation will become \(\frac{-4y-9}{4}-\frac{y}{4}=-1\), which simplifies to \(-y - \frac{9}{4} - \frac{y}{4}=-1\)

Step 3: Solve for y

Now, simply solve the equation for y by multiplying through by 4 and simplifying, it gives \(-4y - 9 - y = -4\), which simplifies further to \(-5y = 5\), thereby \(y= -1\)

Step 4: Substitute y into the expression of x

Now substitute \(y = -1\) to the expression obtained in step 1, which gives \(x=-4(-1)-9 = -5\)

Step 5: Express the answers in set notation

The solution set for the system of equations is expressed in set notation as {(x, y)| x = -5 and y = -1}

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations that have common variables. In simpler terms, it's like having several related mathematical sentences, and you're looking for numbers that satisfy all these sentences simultaneously. Think of a system like two puzzle pieces that fit perfectly together where the edge is common, representing the shared solution points for the equations involved.

In our example system:

• First equation: \( \frac{x}{4} - \frac{y}{4} = -1 \)
• Second equation: \( x + 4y = -9 \)

The goal is to find the values of \( x \) and \( y \) that make both equations true at the same time. When solving such a system, you can use methods like substitution or elimination. In our particular case, we are using the substitution method.

Solving Equations with Substitution

The substitution method involves solving one of the equations for one variable and then substituting that solution into the other equation. In our example, we start with the second equation \( x + 4y = -9 \). We rearrange it to express \( x \) in terms of \( y \):

• Rearrange to get: \( x = -4y - 9 \)

Next, we take this expression for \( x \) and substitute it into the first equation, which allows us to solve for \( y \) alone. The substituted equation becomes \( \frac{-4y-9}{4} - \frac{y}{4} = -1 \).

By solving this equation:

• First clear the fractions by multiplying every term by 4: \( -4y - 9 - y = -4 \)
• Simplify to find \( y = -1 \)

Finally, we substitute \( y = -1 \) back into our rearranged equation for \( x \), ending with \( x = -5 \). Thus, substitution reveals the solution to the system.

Expressing Solutions in Set Notation

Set notation is a fantastic way to represent solution sets because it clearly shows the values that satisfy all the equations in a system. In set notation, we list all solutions that fit the conditions given in curly braces. It's like declaring your puzzle solution is complete.

After solving the system of equations, we discovered that \( x = -5 \) and \( y = -1 \). In set notation, we express this solution as:

• \( \{(x, y) | x = -5, y = -1\} \)

This notation indicates the set of all ordered pairs \( (x, y) \) that are solutions to the system. In our case, there’s just one such pair, showing that this system has a unique solution.