Problem 10 - Algebra 1: Concepts and Skills

Question

Solve the equation. \(3 r-r+15=41\)

Step-by-Step Solution

Answer

The solution to the equation \(3r - r + 15 = 41\) is \(r = 13\).

Step 1: Combine Like Terms on the Left Side of the Equation

Start by combining like terms on the left side. We have \(3r - r\), which simplifies to \(2r\). The equation now is \(2r + 15 = 41\).

Step 2: Subtract 15 from both sides

Next, we'll subtract 15 from both sides of the equation to isolate the \(r\) term. This results in the equation \(2r = 41 - 15\), which simplifies to \(2r = 26\).

Step 3: Divide by 2

Finally, divide both sides of the equation by 2 to solve for \(r\). This gives us \(r = 26 ÷ 2\), which further simplifies to \(r = 13\).

Key Concepts

Combining Like Terms

The concept of combining like terms is a fundamental aspect of solving linear equations. It helps simplify equations, making them easier to manage and solve. In any algebraic expression, "like terms" are terms that have the same variables raised to the same powers. In simple terms, they are terms that look similar in structure. For example, consider terms like \(3r\) and \(-r\). They are like terms because both involve the variable \(r\).
When solving an equation, begin by identifying and combining these similar terms. In the original exercise, the equation \(3r - r + 15 = 41\) contains like terms \(3r\) and \(-r\). Combine these to simplify the expression. Here’s how it works:

• Combine \(3r - r\): you subtract the coefficients of \(r\), resulting in \(2r\).
• The simplified equation becomes \(2r + 15 = 41\).

Combining like terms is a critical step in simplifying and solving equations effectively.

Isolating Variables

Isolating variables is the process of getting the variable of interest on one side of the equation by itself. This is a critical step in solving equations, allowing you to determine the value of the variable. Once the like terms are combined and the equation is simpler, it's time to isolate the variable.
Take the equation, \(2r + 15 = 41\) after simplifying. To isolate \(r\), you need to eliminate any numbers added or subtracted to it. Here's how it works:

• Subtract 15 from both sides of the equation: \(2r + 15 - 15 = 41 - 15\).
• This simplifies to \(2r = 26\).

The equation now is clearer, with \(r\) set for further simplification. Once the variable is isolated, you can easily find its value by further simplification.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations using various algebraic rules. This includes techniques like combining like terms, isolating variables, and applying operations equally to both sides of an equation. Once you have isolated the variable and simplified the equation as \(2r = 26\), it's time for final steps through algebraic manipulation.
In this context, the appropriate manipulation is to divide both sides by the coefficient attached to the variable \(r\), which in this case is 2. Here's how you proceed:

• Divide each side of the equation by 2: \(\frac{2r}{2} = \frac{26}{2}\).
• This gives \(r = 13\).

Through careful algebraic manipulation, you have arrived at the solution. These methods allow for efficient and reliable solving of linear equations, developing problem-solving skills essential in mathematics.