Problem 56 - Understanding Elementary Algebra with Geometry

Question

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 0.2 x+0.02 y &=8 \\ 4 x-y &=20 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \(x = 30\) and \(y = 100\).

Step 1 - Simplify the Equations

First, let's simplify the given equations for easier handling. Start by multiplying the first equation by 100 to eliminate decimals. The first equation becomes: \(20x + 2y = 800\). The second equation remains the same: \(4x - y = 20\).

Step 2 - Solve for y in the Second Equation

Solve the second equation for \(y\): \(4x - y = 20\) Add \(y\) to both sides: \(4x = y + 20\) Subtract 20 from both sides: \(y = 4x - 20\).

Step 3 - Substitute y in the First Equation

Substitute \(y\) from Step 2 into the first equation: \(20x + 2(4x - 20) = 800\) Distribute the 2: \(20x + 8x - 40 = 800\)

Step 4 - Combine Like Terms

Combine like terms to simplify: \(28x - 40 = 800\) Add 40 to both sides: \(28x = 840\).

Step 5 - Solve for x

Divide both sides by 28: \(x = 30\).

Step 6 - Solve for y

Substitute \(x = 30\) back into the expression for \(y\) from Step 2: \(y = 4(30) - 20\) This simplifies to: \(y = 120 - 20 = 100\).

Key Concepts

Algebraic Methods

When solving systems of equations, algebraic methods are invaluable. These techniques include substitution, elimination, and matrix methods. Each method enables us to find the values of the variables in the equations. By understanding these tactics, tackling complex problems becomes much more manageable. For example, in the given exercise, we chose the substitution method, which is particularly useful when one equation is already solved for one of the variables, or can easily be manipulated to be so.

Equation Simplification

Simplifying equations is a critical step in algebra. It involves transforming the given equations into simpler forms that are easier to work with. In our exercise, the first equation was simplified by eliminating the decimal. Multiply each term by 100 to achieve this:

• Original: 0.2x + 0.02y = 8
• Simplified: 20x + 2y = 800

This makes the equation more manageable. Simplifying doesn't change the equation's meaning, but it does make the next steps more intuitive.

Substitution Method

The substitution method involves solving one of the equations for one variable and then substituting this expression into the other equation. This allows you to solve for one variable first and then the other. In our example, we first solved the second equation for \(y\): \[4x - y = 20\] becomes \[y = 4x - 20\]. Next, we substituted \(y\) into the first equation: \[20x + 2(4x - 20) = 800\]. This makes the complex system simpler to manage, resulting in a single-variable equation that can easily be solved.

Linear Equations

Linear equations are equations where the variables are raised to the power of one. For example, both equations in our system are linear:

• 20x + 2y = 800
• 4x - y = 20

These types of equations graph as straight lines on a coordinate plane. Solving a system of linear equations involves finding the intersection point of these lines, which represents the solution to the system. Understanding linear equations is fundamental to solving more complex systems and is a key concept in algebra.