Problem 46 - College Algebra Essentials

Question

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of three times a first number and twice a second number is \(8 .\) If the second number is subtracted from twice the first number, the result is \(3 .\) Find the numbers.

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 2\) and \(y = 1\).

Step 1: Formulate the system of equations

From the conditions given, we have two equations:\n1. Three times first number plus twice second is 8, which translates to \(3x + 2y = 8\)\n2. Second number subtracted from twice first number gives 3, which translates to \(2x - y = 3\)

Step 2: Solve the system of equations

We could use substitution method to solve the equation. First, we could rearrange the second equation to find y in terms of x. This gives us: \(y = 2x - 3\). Next, we substitute y in first equation, which gives us: \(3x + 2(2x - 3) = 8\). Solving this equation for x, we get, \(x = 2\)

Step 3: Find the second number

After determining x, we can substitute x in the second equation to find the value of y: \(y = 2*2 - 3 = 1\)

Key Concepts

Algebraic Equations

Algebraic equations are a foundational component in mathematics, forming the basis for many complex operations in higher-level math. They constitute expressions with one or more unknown variables, like an enigma waiting to be solved. In essence, the equation represents a scale, balancing two expressions; it is only considered 'solved' when the value of the unknown variables makes both sides equal.

Take, for instance, the equation presented in the exercise \(3x + 2y = 8\). Here, \(x\) and \(y\) are the unknowns—numbers we’re trying to find. The numbers 3 and 2 are the coefficients, representing the 'weight' of each variable in the equation. The solution to this kind of equation requires an intricate dance of arithmetic operations, aimed at isolating and determining the value of each unknown.

Substitution Method

The substitution method is a powerful and intuitive algebraic tool for unraveling the mysteries of variables within a system of equations. Like a detective who substitutes one clue for another to unravel a mystery, this method involves taking the solution from one equation and substituting it into another, thereby reducing the system to a single equation about one variable.

When we look at our exercise, after rearranging the second equation, we get \(y = 2x - 3\). This new piece of information is substituted into the first equation to eliminate \(y\), creating an equation with only one unknown, \(x\). Now, we're effectively solving the simpler equation \(3x + 2(2x - 3) = 8\) to find the value of \(x\), and this vital step brings us closer to the solution. Using substitution smoothly transforms a complex system into something much more manageable.

Solving Linear Systems

Solving linear systems is akin to piecing together a puzzle where each piece must fit perfectly to reveal the full picture. A linear system consists of two or more equations with the same variables, and the objective is to find a common solution for these variables that satisfies all equations simultaneously.

In our textbook exercise, we are presented with a linear system with two equations: \(3x + 2y = 8\) and \(2x - y = 3\). The strategy to tackle such systems can vary—from graphing and finding the intersection point, to algebraic methods like substitution (used in this exercise), or even elimination, where one adds or subtracts equations to cancel out a variable. The joy of solving a linear system comes when you land on the solution that clicks, and both equations are contentedly true, much like solving a whodunit train-track mystery where all clues point to just one perpetrator.