Problem 1 - Technical Mathematics with Calculus

Question

Solve simultaneously. Check some by calculator. $$\begin{aligned} &\frac{x}{5}+\frac{y}{6}=18\\\ &\frac{x}{2}-\frac{y}{4}=21 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system is x = 93.75 and y = 103.5.

Step 1: Multiply equations to eliminate denominators

Multiply both sides of the first equation by 30, the least common multiple of 5 and 6, to eliminate the denominators. Do the same for the second equation using the least common multiple of 2 and 4, which is 4.

Step 2: Rewrite the equations

After multiplying, the equations become: 1) 6x + 5y = 10802) 2x - y = 84

Step 3: Multiply equations for elimination method

Multiply equation 2) by 5 so that y coefficients are opposites: 2) 10x - 5y = 420

Step 4: Add the equations to eliminate y

Add the new equation 2) to equation 1) to eliminate y: 6x + 5y + 10x - 5y = 1080 + 420

Step 5: Solve for x

After combining like terms, we have:16x = 1500Now, divide both sides by 16 to solve for x:x = 93.75

Step 6: Substitute x into one of the original equations

Use equation 2) to solve for y:2x - y = 84Substitute x = 93.75 into the equation to find y.

Step 7: Solve for y

After substitution, the equation becomes:2(93.75) - y = 84This simplifies to:187.5 - y = 84Subtracting 187.5 from both sides gives:-y = -103.5Multiplying through by -1:y = 103.5

Step 8: Check solution using a calculator

Use a calculator to plug the values of x = 93.75 and y = 103.5 back into the original equations to ensure they both hold true.

Key Concepts

Elimination Method

The elimination method is a powerful tool for solving simultaneous linear equations—those in which two or more equations are solved together to find common solutions for the variables involved. When presented with a system like \[\frac{x}{5} + \frac{y}{6} = 18 \] and \[\frac{x}{2} - \frac{y}{4} = 21\], the goal is to manipulate the equations so a variable is eliminated, allowing for easier solution.

The first step in the elimination process often involves clearing fractions by multiplying each term by a common denominator. In our example, multiplying the first equation by 30 and the second by 4 does the trick. This forms a new system of equations without fractions, which is easier to work with:\[6x + 5y = 1080 \] and \[2x - y = 84\].

Next, we want to create opposite coefficients for either the x's or the y's. This is done by strategically multiplying the equations. Once the coefficients are opposites, we add or subtract the equations to eliminate one variable. In our exercise, the second equation is multiplied by 5, yielding \[10x - 5y = 420\], which perfectly sets up y for elimination when added to the first equation.

After eliminating y, we solve for x, and then back-substitute to find y. This method allows for a step-by-step solution, ensuring that each variable can be calculated with precision.

Linear Equations

Linear equations, a central topic in algebra, are expressions that describe a straight line when plotted on a graph. These equations have the general form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. The power of linear equations lies in their simplicity and their abundance in real-world applications.

When dealing with simultaneous linear equations, you have multiple equations that must be satisfied at the same time, finding the point of intersection in a graph representing the solution.For complex cases, like those with fractions or multiple variables, it's common practice to manipulate the equations to simplify them, often seeking a form without denominators that are easier to work with.

In our example, transforming \(\frac{x}{5} + \frac{y}{6} = 18\) and \(\frac{x}{2} - \frac{y}{4} = 21\) into a simplified form leads to an approachable problem that hinges on understanding these fundamental properties of linear equations.

Algebraic Methods

Algebraic methods encompass a broad range of techniques used to solve equations. These methods are the backbone of algebra and include operations such as addition, subtraction, multiplication, division, and the use of exponents and roots to manipulate and solve equations.

In solving simultaneous equations, algebraic methods offer a structured approach: identify the least common multiples, employ the elimination method, and perform substitutions to find the values of unknowns. These techniques reveal the interconnected nature of algebraic equations and demonstrate how changing one part of an equation affects the whole.

For the exercise at hand, each step represents an application of algebraic methods, from eliminating denominators to simplifying complex fractions. Mastering these methods not only helps in finding the solution but also develops a keen understanding of how to approach similar problems. The step-by-step logic applied ensures that even when equations initially appear daunting, they can be broken down into manageable parts, leading to a clear and dependable solution.