Problem 21 - Introductory Algebra for College Students

Question

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 3 x=2 y+7 \\ 5 x=2 y+13 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \({(x, y) | x = 3, y = 1}\).

Step 1: Initial System of Equations

The system of linear equations given: \(3x = 2y + 7\) and \(5x = 2y + 13\).

Step 2: Applying the Addition Method

Subtract the second equation from the first equation. This will cancel out the term with \(y\) as the coefficient of \(y\) is the same in both equations, and makes it easy to isolate \(x\). Completing the calculation, we get: \(-2x = -6\).

Step 3: Calculate the Value of x

Solving for \(x\) we see \(x = -6/(-2) = 3\).

Step 4: Substitute \(x\) into one of the equations

Substitute \(x = 3\) into the first equation: \(3*3 = 2y + 7\) gives \(9 = 2y + 7\).

Step 5: Calculate the Value of y

Rearrange to get \(2y = 9 - 7 = 2\), then divide by 2 to find, \(y = 2/2 = 1\).

Step 6: Write the solution set

The solution to this system of equations is the coordinate point (3,1), expressed in set notation as \({(x, y) | x = 3, y = 1}\).

Key Concepts

Understanding the Addition Method

The addition method, also known as the elimination method, is a reliable way to solve systems of linear equations. It involves adding or subtracting equations to cancel out one of the variables. This helps in finding the value of the remaining variable easily.
To apply the addition method effectively, follow these steps:

• Quickly check for coefficients: Look for coefficients of the same variable that are either identical or additive inverses.
• Add or subtract the equations: Depending on the coefficients, either add or subtract the whole equation to eliminate one variable.
• Solve for the remaining variable: Once one variable is eliminated, solve for the other variable as shown in the step-by-step solution.

By canceling out variables methodically, the addition method streamlines the process of arriving at a solution, making it a favorite for many students dealing with linear systems.

Determining the Solution Set

Once the addition method provides you with the values of the variables, you can establish the solution set. The solution set is a representation of all answers to the system of equations. It is typically shown as an ordered pair indicating the point where the two lines intersect.
For the given system, where \(x = 3\) and \(y = 1\), the coordinates are shown as \( (3, 1) \).
In set notation, this is expressed as: \[ \{(x, y) | x = 3, y = 1\} \] This tells us that the only solution to the system is at the specific point (3, 1) on the Cartesian plane. This method of expressing solutions helps verify correctness when checking work.

Understanding Linear Equations

Linear equations form the backbone of many mathematical concepts. Each equation represents a straight line when graphed on a coordinate plane. These equations usually appear in the form \(ax + by = c\), where \(a, b,\) and \(c\) are constants.

Some important characteristics of linear equations include:

• Their graph is always a straight line.
• The solution to a system of linear equations will be the point where these lines intersect, if they do intersect.
• If two linear equations have the same slope but different y-intercepts, they'll be parallel and have no solutions.
• If they have the same slope and the same y-intercept, they'll be identical, indicating infinite solutions.

Understanding these principles makes it easier to grasp how linear equations interact and why methods like the addition method work effectively. They give a straightforward plan to approach and solve these types of problems.