Problem 6 - 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. $$\left\\{\begin{array}{l}x+2 y=5 \\\2 x-y=-15\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \{(-5, 5)\}

Step 1: Isolate a variable in the first equation

From the equation \(x + 2y = 5\), let's isolate variable \(x\). By rearranging, \(x = 5 - 2y\)

Step 2: Substitute the isolated variable in the second equation

Now substitute \(x\) in the second equation \(2x - y = -15\), we get \(2(5 - 2y) - y = -15\). After simplifying, it gives \(-5y = -25\). So, \(y = 5\)

Step 3: Substituting \(y\) in the isolated \(x\)

Take \(y = 5\) and substitute back in \( x = 5 - 2y \). It gives \(x = -5\)

Step 4: Expressing the answer in set notation

The answer is given in set notation as \{(x,y)\} where \(x = -5\) and \(y = 5\), So, \{(-5, 5)\}

Key Concepts

System of Equations

When dealing with a **system of equations**, you work with two or more equations that include the same variables. In our case, we have two linear equations involving variables \(x\) and \(y\). Solving such a system involves finding a set of values for these variables that satisfy all the equations at once.

• A system can have a unique solution, meaning there is a distinct pair of \(x\) and \(y\) that works.
• It might have no solution if the lines represented by the equations are parallel.
• Sometimes, a system could even have an infinite number of solutions if the equations describe the same line.

In this exercise, our system is:\[\begin{align*}& x + 2y = 5 \& 2x - y = -15\end{align*}\]Finding the solution involves intersecting these two equations, essentially finding where the lines meet.

Solution Set

The **solution set** for a system of equations represents the values of the variables that satisfy all equations in the system. For linear equations, these solutions often come in the form of points where the lines intersect.
To express the solution set:

• We use set notation which formally defines the solution as combinations of \(x\) and \(y\).
• In our problem, the solution is \((-5, 5)\), where the point \(x = -5\) and \(y = 5\) satisfies both equations.

If there's no solution, we denote it as an empty set, \(\{\}\). If there are infinitely many solutions, such as with overlapping lines, this is described in the context of the problem.

Introductory Algebra

**Introductory algebra** is an essential building block where you begin learning how to manipulate equations and solve for unknowns. It involves understanding how variables work and learning the rules to simplify and solve equations.
In our exercise, basics of algebra come into play like:

• Rearranging equations to isolate variables.
• Substituting values to simplify calculations.
• Understanding how combining different pieces of information can lead to solving a problem.

Mastering these steps is crucial as it allows you to tackle more complex problems in mathematics.

Solving Linear Equations

**Solving linear equations** is about finding the values of variables that make the equation true. In the context of a system of equations, you often solve one equation to express one variable in terms of another, then substitute back to find the other variable.
Here's a breakdown of the steps we used with the substitution method in our example:

• Step 1: Isolate one variable (here, \(x\)) in one of the equations.
• Step 2: Substitute this variable into the second equation to find the value of the other variable (\(y\)).
• Step 3: Once the second variable is found, substitute back to get the first variable.
• Step 4: Confirm that these values make both equations true and write your solution in set notation.

This systematic approach ensures that you capture the full solution effectively.