Problem 14 - 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. $$\left\\{\begin{array}{l} 2 x-5 y=-1 \\ 3 x+y=7 \end{array}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Rearrange the Equations

First, rearrange the equations to isolate the variable y in the second equation. The system becomes: \[ \begin{cases} 2x - 5y = -1 \ y = 7 - 3x\end{cases}\]

Step 2: Substitute y in the First Equation

Replace y in the first equation with the expression from the second equation: \[ 2x - 5(7 - 3x) = -1 \] Simplify it to find the value of x: \[ 2x - 35 + 15x = -1 \]\[ 17x = 34 \]\[ x = 2 \]

Step 3: Find the Value of y

Substitute x = 2 into the second equation: \[ y = 7 - 3*2 \]\[ y = 1 \]

Step 4: Write the Solution in Set Notation

The solution set for the system of equations is composed of ordered pairs (x, y) that satisfy both equations. In this case, there is only one solution: \[ {(2, 1)} \]

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. In algebra, this typically involves linear equations, but systems can also involve nonlinear equations. The main goal when solving a system of equations is to find values for the unknowns that satisfy all of the given equations simultaneously.

To solve systems of equations, several methods can be used including:

• Graphical Method: Visually identifying the intersection points of the graphs
• Substitution: Solving one equation for one variable and substituting into another
• Addition (or Elimination) Method: Adding or subtracting equations to eliminate a variable

In the given exercise, we use the addition method to find a common solution for the variables x and y. If the equations in a system are consistent, they will intersect at least at one point, representing one or more solutions. If not, they may be parallel, indicating no solution, or identical, leading to infinite solutions.

Solution Set

The solution set for a system of equations includes all the possible values of the variables that satisfy each equation in the system. With linear equations, the solution set is represented as an ordered pair (x, y) in two-dimensional cases. This set represents the exact point where the equations intersect in the Cartesian plane.

• Unique Solution: If the system has only one solution, the two lines intersect at a single point, as with our determined solution (2, 1).
• No Solution: If the lines are parallel, they will never intersect, implying no common solution.
• Infinite Solutions: When two equations represent the same line, all points on the line are solutions.

Set notation is often used to concisely represent the solution set. For example, \({ (2, 1) } \) indicates that when x equals 2, and y equals 1, both equations are satisfied.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. They form the basis of equations and represent quantities in algebra. In a system of equations, each equation is an algebraic expression set equal to another expression or a constant.

• Consider the expression in the exercise: \( 2x - 5y = -1 \)
• And \( 3x + y = 7 \)

Each of these consists of:

• Variables: Symbols like x and y that hold the place of numbers.
• Constants: Specific, unchanging values like -1 and 7.
• Coefficients: Numbers multiplying the variables, such as 2, -5, and 3.
• Operators: Mathematical actions like addition (+) and subtraction (-).

Solving these requires manipulating the expressions using algebraic rules to isolate the variables, as done in the steps of the solution process to solve for x and y.