Problem 27 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ r-2[1-3(2 r+4)]=61 $$

Step-by-Step Solution

Answer

The solution to the equation is \( r = 3 \).

Step 1: Distribute Inside Brackets

First, let's simplify the expression inside the brackets. Start with the innermost bracket: \( 3(2r + 4) \).Distribute 3 into the terms inside the bracket:\( 3 \times 2r = 6r \) and \( 3 \times 4 = 12 \).So, \( 3(2r + 4) = 6r + 12 \).

Step 2: Simplify Inside the Square Brackets

Next, simplify inside the square brackets:\( 1 - (6r + 12) = 1 - 6r - 12 = -6r - 11 \).

Step 3: Eliminate the Square Brackets

Now substitute the simplified expression back into the equation:\( r - 2(-6r - 11) = 61 \).Distribute \(-2\) to the terms inside the bracket:\( -2 \times -6r = 12r \) and \( -2 \times -11 = 22 \).So, the equation becomes: \( r + 12r + 22 = 61 \).

Step 4: Combine Like Terms

Combine the terms with \(r\):\( r + 12r = 13r \).So now the equation is:\( 13r + 22 = 61 \).

Step 5: Isolate r

To solve for \(r\), first subtract 22 from both sides:\( 13r + 22 - 22 = 61 - 22 \).Simplifying this gives:\( 13r = 39 \).

Step 6: Solve for r

Finally, divide both sides by 13 to solve for \(r\):\( 13r/13 = 39/13 \).This simplifies to:\( r = 3 \).

Key Concepts

Solving Linear Equations

Linear equations are mathematical statements expressing equality involving variables raised to the power of one. Solving them involves finding the value of the variable that makes the equation true. Imagine you have a simple balance scale. Solving a linear equation is like reorganizing the weights on the scale until both sides are equal.
Linear equations can appear complex, but you can apply certain steps and operations to isolate the variable. For example, in the equation \( r - 2[1-3(2r+4)] = 61 \), we aim to isolate \( r \).
To achieve this, we can use basic arithmetic operations like addition, subtraction, multiplication, or division. Solving these equations often involves several key steps that build on each other.

Distributive Property

The distributive property is a useful tool when solving equations involving parentheses. It allows you to remove the parentheses by distributing a multiplier across terms within a bracket. For instance, if you have \( a(b + c) \), you can multiply \( a \) by \( b \) and then \( a \) by \( c \), giving you \( ab + ac \).
In the original exercise, you have an expression within brackets: \( 3(2r+4) \). To simplify, distribute the 3 to both \( 2r \) and \( 4 \), resulting in \( 6r + 12 \). This step simplifies the equation and makes it easier to continue solving.
Later, you encounter another set of brackets with \( -2(-6r - 11) \). Applying the distributive property again results in a new expression, making it simpler to address the variable terms.

Combining Like Terms

Combining like terms involves merging terms in an equation that have the same variable or can be directly added together. This simplification helps to bring equations to a form that's easier to solve. Look for terms that share the same variables and exponents, then sum or subtract their coefficients.
In the exercise, once the distributive property is applied and you rewrite the equation as \( r + 12r + 22 = 61 \), you should notice that \( r \) and \( 12r \) are like terms. Adding them together simplifies the equation to \( 13r + 22 = 61 \).
By combining these terms, you reduce the complexity of the equation, moving it closer to isolating the variable, \( r \).

Isolation of Variables

Isolation of variables is the method used to solve an equation for a specific variable. The goal is to have the variable alone on one side of the equation, with all other terms on the opposite side. This step is crucial for finding the value of the variable.To achieve isolation, you manipulate the equation using inverse operations. For example, in our original equation \( 13r + 22 = 61 \), we need to isolate \( r \).
Start by removing the constant term \( 22 \) from the left side by subtracting it from both sides, giving you \( 13r = 39 \). Next, divide both sides by 13 to solve for \( r \), resulting in \( r = 3 \).By systematically applying arithmetic operations, you can isolate the variable and solve the equation effectively.