Problem 58 - Algebra 1

Question

Round the result to the nearest hundredth. Check the rounded solution. $$2.6(4.2 y-6.5)=-7.1 y-2.8$$

Step-by-Step Solution

Answer

\nIndeed, the short answer for \(y\) is \(0.78\), rounded to the nearest hundredth.

Step 1: Distribute Multiplication

Distribute 2.6 across \(4.2y - 6.5\) to remove the parentheses: \(2.6 \times 4.2y - 2.6 \times 6.5 = -7.1y - 2.8\)

Step 2: Simplify Both Sides of the Equation

After distributing, you get \(10.92y - 16.9 = -7.1y - 2.8\). On one side of the equation, add \(7.1y\) to consolidate the \(y\) terms: \(10.92y + 7.1y = -2.8 + 16.9\)

Step 3: Solve for y

After adding like terms, you get \(18.02y = 14.1\). Now, divide both sides by \(18.02\) to solve for \(y\): \(y = 14.1 / 18.02\)

Step 4: Round to the Nearest Hundredth

When you divide \(14.1 / 18.02\), you get \(y = 0.7819093198999439\), which you need to round to the nearest hundredth. This results in \(y = 0.78\)

Step 5: Check the Solution

To ensure the correctness of the solution, plug \(0.78\) back into the original equation for \(y\) and verify that both sides of the equation match. That is, \(2.6(4.2 \times 0.78 - 6.5)\) should equal \(-7.1 \times 0.78 - 2.8\)

Key Concepts

Distributive Property

The distributive property is a key algebraic property used to simplify expressions. It involves multiplying a single term by each term inside a set of parentheses. This property can be represented by the formula:

• a (b + c) = ab + ac

In our exercise, we see this property in action with the expression \(2.6(4.2y - 6.5)\). By applying the distributive property, we multiply 2.6 by each term inside the parentheses. This transforms the expression into:

• \(2.6 \times 4.2y - 2.6 \times 6.5\)

This crucial step helps us to remove the parentheses and prepare the equation for further simplification.

Simplifying Expressions

Once the distributive property is applied, the next step is simplifying the expression. Simplifying involves reducing the equation to its most basic form without changing its value. For the given problem, after applying the distributive property, we have:

• \(10.92y - 16.9 = -7.1y - 2.8\)

To simplify this, we need to group and combine like terms. Add \(7.1y\) to both sides to group the \(y\)-terms:

• \(10.92y + 7.1y = -2.8 + 16.9\)

This gives us \(18.02y = 14.1\). Simplifying expressions reduces complexity, making it easier to solve for variables later.

Rounding Numbers

Rounding involves adjusting a number to a specified degree of accuracy or precision, often to make calculations simpler or the results more understandable. In this exercise, after solving for \(y\), we obtained a result of \(0.7819093198999439\). However, since the question asks to round to the nearest hundredth, let's explore how to do this.

• Identify the digit in the hundredths place, which is the second digit after the decimal: in \(0.7819093198999439\), it's 8.
• Look at the digit immediately to the right (the thousandths place). Here, it's 1.
• Since this digit is less than 5, we do not round up the digit in the hundredths place, leaving it as is.

Thus, the rounded value is \(y = 0.78\). Rounding ensures that answers meet specified precision requirements.

Solving for Variables

Solving for a variable means isolating it on one side of the equation to find its value. Once we've simplified the expression down to \(18.02y = 14.1\), our goal is to solve for \(y\).

• Divide both sides of the equation by 18.02 to isolate \(y\).

This results in:

• \(y = \frac{14.1}{18.02}\)

Carrying out this division gives \(y = 0.7819093198999439\), which we previously rounded to \(y = 0.78\). Solving for variables is a fundamental skill in algebra, allowing us to find unknowns and understand their relationships within equations.