Problem 27 - 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. $$\left\\{\begin{array}{l}y=\frac{1}{3} x+\frac{2}{3} \\\y=\frac{5}{7} x-2\end{array}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Setting the Expressions Equal to Each Other

Since both of the equations in the system are already solved in terms of \(y\), you can set these two expressions equal to each other: \[ \frac{1}{3} x+\frac{2}{3} = \frac{5}{7} x-2 \]

Step 2: Solve for \(x\)

Now manipulate the equation to solve for \(x\). First, multiply all terms by 21 to eliminate the fractions and combine like-terms: \[ 7x + 14 = 15x - 42 \]. Then, subtract 7x from both sides so as to gather all \(x\)'s on one side: \[14 = 8x - 42\]. Afterwards, add 42 to both sides: \[ 56 = 8x \]. Finally, divide both sides by 8, obtaining \(x = 7\).

Step 3: Solve for \(y\)

Then substitute \(x = 7\) into both original equations and solve for \(y\). You should find that for both equations, when \(x = 7\), \(y = 3\). Thus the solution to the system is \(x = 7, y = 3\) or \( (7,3) \).

Step 4: Expressing Solution in set notation

In set notation, any solution pair \((x,y)\) can be written as a set, \{(x,y)\}. Hence, the result is \{(7,3)\}.

Key Concepts

Systems of Equations

When you encounter a system of equations, you're looking at two or more equations that share the same variables. The goal with these systems is to find a common solution that satisfies all the equations involved. For linear systems like the one in our exercise, each equation represents a line on a graph.

• When the lines intersect at a single point, there’s one unique solution.
• When the lines are parallel and distinct, they never meet, offering no solution.
• If the lines overlap completely, they have infinite solutions.

The substitution method, as used here, involves solving one of the equations for a single variable. Then, you substitute that expression into the other equation. This results in one equation with one variable, which is easier to solve.

Set Notation

Set notation is a way to clearly communicate the solution set of a mathematical problem. This is particularly useful when dealing with systems of equations, as it offers a concise method to display the solutions.
In our problem, we found the solution to be the point \( (7, 3) \). To express this in set notation, we write it as \( \{(7, 3)\} \). This notation signifies that the solution is a set containing one ordered pair, displaying both values for \( x \) and \( y \).

• Use curly braces \( \{ \} \) to encompass the set.
• The pair \( (x, y) \) indicates the coordinates of the solution.

Set notation neatly encapsulates the result, making it clear that the solution satisfies the given equations.

Solving Linear Equations

Solving linear equations is a foundational skill in mathematics. When using the substitution method, you want to isolate one variable, then substitute its expression back into the other equation.
In our example, both equations were already set equal to \( y \), so we simply equated them: \[ \frac{1}{3} x + \frac{2}{3} = \frac{5}{7} x - 2 \].
After clearing fractions and simplifying, we solved for \( x \) by performing basic operations like adding, subtracting, dividing, and multiplying. Follow these steps:

• Combine like terms on each side of the equation.
• Move variables to one side and constants to the other to solve for one variable.
• Next, substitute back to get the value of the other variable.

This consistent process will guide you through solving any linear equation, ensuring accuracy and clarity.