Problem 50 - Introductory Algebra for College Students

Question

Solve each equation and check your proposed solution in Exercises. $$0.15 y-0.1=2.5 y-1.04$$

Step-by-Step Solution

Answer

The solution of the linear equation \(0.15y - 0.1 = 2.5y - 1.04\) is \(y \approx 0.485\).

Step 1: Combine like terms on each side

Rearrange the terms to group 'y' terms together on one side and the constant terms on the other. This results in: \(2.5y - 0.15y = 1.04 + 0.1\).

Step 2: Simplify both sides of the equation

On the left-hand side, subtract the 'y' terms: \(2.5y - 0.15y = 2.35y\). On the right-hand side, add the constants: \(1.04 + 0.1 = 1.14\). Now the equation looks like this: \(2.35y = 1.14\).

Step 3: Solve for 'y'

To solve for 'y', the coefficient of 'y' needs to be made 1. This can be done by dividing all terms of the equation by the coefficient 2.35 to isolate the variable 'y' on one side. Thus, 'y' equals to: \(y = 1.14 / 2.35\).\(

Step 4: Simplify the solution

Perform the division to get the value of 'y'. \('y' approximately equals to 0.485.\)

Step 5: Check the solution

Substitute 'y' into the original equation to validate the solution. If both sides of the equation are equal, then the solution is correct. Substituting 'y' into original equation yields: \(0.15 * 0.485 - 0.1 = 2.5 * 0.485 - 1.04\), which simplifies to: \(-0.02725 = -0.02725\). Thus, the solution is correct, as both sides of the equation are equal.

Key Concepts

Combining Like Terms

When we look at an algebraic expression or equation, we often find terms that are similar in nature, which means they have the same variable raised to the same power. Combining like terms is a fundamental process in simplifying expressions and solving equations.

For example, in the equation from our exercise, we see terms that contain 'y'. To combine them, we add or subtract the coefficients (the numbers in front of the variables) while keeping the variable part the same. In our step 1, we combined the 'y' terms on one side which resulted in the equation:
\(2.5y - 0.15y = 1.04 + 0.1\).
After combining the like terms, we obtained \(2.35y\) on the left side by simply subtracting the coefficients. Similarly, we combined the constants on the right, resulting in \(1.14\).

This process is crucial because it simplifies the equation, making the next steps clearer and easier to manage.

Isolating Variables

One of the main goals in solving algebraic equations is to isolate the variable we are solving for. Isolating a variable means rearranging the equation so that the variable appears on one side by itself. This is achieved through various algebraic operations such as addition, subtraction, multiplication, and division, applied to both sides of the equation to maintain equality.

In our problem from step 3, we isolated 'y' by dividing both sides by \(2.35\), the coefficient of 'y':
\(y = \frac{1.14}{2.35}\).
This operation effectively removes the coefficient and leaves the variable 'y' by itself on one side of the equation. Isolating variables is a vital step towards finding the solution to the equation.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them to their simplest form while retaining their value. This may include several steps like combining like terms, using distributive properties, and factoring out common factors.

In our step 2, after combining like terms, we simplified both sides of the equation by performing the arithmetic operations:
\(2.5y - 0.15y = 2.35y\) and \(1.04 + 0.1 = 1.14\).
The simplicity achieved here made it straightforward to move to the next step of isolating the variable. Simplification helps in making complex problems more manageable by breaking them down into simpler parts that can be easily solved.

Checking Solutions

After finding a potential solution to an equation, it's crucial to verify its validity. This process is known as checking the solution. We do this by substituting the value of the variable back into the original equation. If the equation balances (the left side equals the right side), the solution is valid.

In our final step from the problem, we substituted our calculated value of 'y' back into the original equation and verified that both sides of the equation remained equal:
\(0.15 * 0.485 - 0.1 = 2.5 * 0.485 - 1.04\).
This step is essential as it ensures that any potential algebraic errors are caught, and you have the correct answer. Always remember to check your solutions to confirm their accuracy.