Problem 67 - Introductory Algebra for College Students

Question

In Exercises \(61-68,\) solve each system or state that the system is inconsistent or dependent. $$\left\\{\begin{array}{l} \frac{x}{2}=\frac{y+8}{3} \\ \frac{x+2}{2}=\frac{y+11}{3} \end{array}\right.$$

Step-by-Step Solution

Answer

The system of equations is dependent with infinitely many solutions.

Step 1: Reformulate Equation 1

Multiply both sides of the first equation by 2 and 3 to eliminate the denominator such that it becomes a simple linear equation of the format y = mx + b. This manipulation results in: \(3x = 2y + 16\). Further simplify the equation to get \(2y = 3x - 16\) then \(y = 1.5x - 8\)

Step 2: Reformulate Equation 2

Similar to the first step, multiply both sides of the second equation by 2 and 3 to eliminate the denominator such that it becomes a simple linear equation of the format y = mx + b. This manipulation results in: \(3x + 6 = 2y + 22\). Further simplify the equation to get \(2y = 3x - 16\) then \(y = 1.5x - 8\)

Step 3: Compare the two equations

From the above steps, the equations for the first and second lines are the same, \(y = 1.5x - 8\). Therefore, these equations define the same line in the plane and they intersect at every point on the line, which means they have infinitely many solutions

Key Concepts

Understanding System of Equations

In mathematics, a system of equations is a set of two or more equations with the same variables. For example, if you have equations with variables x and y, these are connected in a way that solutions for these variables need to satisfy all equations simultaneously. A system of equations can be represented in different manners, but often they are shown as multiple mathematical statements grouped together, like in brackets.
Solving a system involves finding the set of values for your variables that make each of the equations true, usually involving techniques like substitution or elimination.

• **Types of Solutions:** There are typically three possible outcomes when solving such systems - one solution (the point where all lines intersect), no solution (parallel lines never meeting), or infinitely many solutions (the lines overlap entirely).
• **Applications:** Systems of equations are used in various real-life contexts, such as finding the intersection of different linear trends or resolving multiple criteria in engineering models.

Delving into Linear Equations

Linear equations are a fundamental concept in algebra and represent straight lines when graphed on a coordinate plane. Each equation in the form of y = mx + b, where m represents the slope and b the y-intercept, showcases a linear relationship between variables x and y.
Linear equations are easy to deal with because their solutions can be quickly represented graphically or calculated using basic algebra.

• **Slope and Intercept:** These two components define the line. Slope (m) determines the line's direction and steepness, while y-intercept (b) is the point where the line crosses the y-axis.
• **Solving Linear Equations:** Solving involves isolating the variable of interest on one side of the equation, often by performing operations like addition, subtraction, multiplication, or division. This is what transforms abstract algebraic expressions into clear and concise solutions.

Identifying Dependent Systems

A dependent system of equations occurs when the equations, while appearing different initially, ultimately represent the same line. When comparing two linear equations and they simplify to an identical expression (e.g., the cases in this exercise), the system is dependent.

A dependent system, also referred to as being 'consistent and dependent,' implies:

• There is an infinite number of solutions as the lines are coincident; they lie on top of one another.
• Graphically, dependent systems display overlapping lines—meaning any point on one line simultaneously satisfies the other.
• Recognizing a dependent system is crucial when solving, as typical methods might not readily show a single solution.

This insight saves time in calculations and provides better intuition about the nature of these kinds of algebraic systems.