Problem 50 - Introductory Algebra for College Students

Question

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{aligned} 2 x+3 y &=7 \\ x &=2 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(\{(2, 1)\}\).

Step 1: Understand the System of Equations

The given system of equations is \(\left\{\begin{aligned} 2x + 3y &=7 \ x &=2 \end{aligned}\right.\)This system contains two equations: the first one is a linear equation in two variables x and y, and the second one is an equation where x is already solved.

Step 2: Substitute x value from second equation into first

The value of x from the second equation can be substituted into the first equation. This will give:\(2*(2) + 3y = 7\), which simplifies to \(4 + 3y = 7\).

Step 3: Solve for y

Subtract 4 from both sides of the equation to isolate y: \(3y = 7 - 4\), which simplifies to \(3y = 3\). Then, divide both sides by 3 to solve for y, which gives \(y = 1\).

Step 4: Express the Solution in Set Notation

The solution is the set of all pairs (x, y) that satisfy both equations. Here, x=2 and y=1 is the solution. Therefore, the solution set in set notation is \(\{(2, 1)\}\).

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves replacing one variable with its equivalent expression from another equation. This method is particularly useful when one of the equations is already solved for a variable. In our exercise, the second equation is already solved for \(x\), making it ideal for substitution.

Here's a quick rundown on how to apply it:

• Find one equation solved for a variable (like \(x = 2\) in this case).
• Substitute this value into the other equation to create an equation with just one variable.
• Solve the simplified equation to find the value of the remaining variable.

This method is straightforward when substitution doesn't make the equation overly complicated. Choosing substitution here helped streamline the process, allowing a quick path to the solution.

Linear Equations

Linear equations are equations of the first degree, meaning they can be graphed as straight lines. In the given system \(2x + 3y = 7\) and \(x = 2\), both are linear equations. The first equation involves two variables, while the second directly gives the value of one variable.

To solve linear equations, it's essential to:

• Understand the coefficients and constants. Coefficients multiply the variables, while constants are standalone numbers.
• Perform operations like addition, subtraction, multiplication, or division uniformly across the equation.
• Graphically, the solution represents the point where the two lines intersect (if they do).

In our solution, isolating \(y\) after substituting \(x\) allowed us to find the intersection of these linear equations, which is \((2, 1)\).

Set Notation

Set notation is a systematic way to express solutions to equations. It presents the solution as a collection of elements within curly brackets. In a system of equations, the solution set contains all ordered pairs \((x, y)\) that satisfy both equations.

To express solutions in set notation:

• Ensure that the values found satisfy both original equations.
• Write the solutions as ordered pairs inside curly braces. For example, \(\{(2, 1)\}\).
• Set notation helps convey whether there are no solutions, one solution, or infinite solutions.

By using set notation, we clearly signify that \((2, 1)\) is the only pair that works for both equations in this system. It's a concise and precise way of stating results.