Problem 27 - Algebra Form and Function

Question

Put the equation in standard form. $$ x+4=3(y-1) $$

Step-by-Step Solution

Answer

Question: Put the equation x + 4 = 3(y - 1) in standard form. Answer: x - 3y = -7

Step 1: Distribute the constant

First, we'll distribute the 3 on the right side of the equation to the terms within the parentheses. $$ x + 4 = 3(y - 1) \\ x + 4 = 3y - 3 $$

Step 2: Isolate variable terms on the left side

Now, let's move the y-term to the left side by subtracting 3y from both sides of the equation: $$ x - 3y + 4 = -3 $$

Step 3: Move the constant to the right side

We'll now move the constant term (+4) to the right side of the equation by subtracting 4 from both sides: $$ x - 3y = -3 - 4 \\ x - 3y = -7 $$ Now, the equation is in the standard form, where A = 1, B = -3, and C = -7. Therefore, the final equation is: $$ x - 3y = -7 $$

Key Concepts

Linear Equations

Linear equations are equations that create a straight line when graphed. They typically involve one or more variables, and the highest power of these variables is one. For example, a simple linear equation looks like this: \( y = mx + b \). This is commonly known as the slope-intercept form, where:

• \( m \) represents the slope of the line
• \( b \) denotes the y-intercept, or where the line crosses the y-axis

However, linear equations can take many forms, not just the slope-intercept form.

When converting equations into standard form, the equation looks like \( Ax + By = C \), where:

• \( A \), \( B \), and \( C \) are constants.
• \( A \) should be a non-negative integer in the final equation.

Understanding these forms allows us to seamlessly shift between representing equations whenever needed.

Distributive Property

The distributive property is a key algebraic property used to simplify expressions and solve equations. It states that multiplying a sum by a number gives the same result as multiplying each addend individually by the number and then adding the products.

In mathematical terms, the distributive property is expressed as \( a(b + c) = ab + ac \). This property is incredibly useful when dealing with expressions inside parentheses.

In our exercise, we initially had \( x + 4 = 3(y-1) \). Using the distributive property, we multiplied 3 by both \( y \) and \(-1\), resulting in \( 3y - 3 \). This step helps break down complex expressions into simpler parts, making it easier to manage and solve the equation.

This property simplifies not only linear equations but also more complex algebraic expressions. Being comfortable with the distributive property is valuable in many areas of mathematics.

Equation Transformation

Equation transformation involves modifying an equation to bring it into a different, often simpler, form while maintaining its equality. This process usually includes several algebraic techniques such as addition, subtraction, multiplication, division, and utilization of properties like the distributive property.

In the context of our exercise, we wanted to bring the equation \( x + 4 = 3(y-1) \) into standard form. To achieve that, several transformations were necessary:

• First, applying the distributive property to remove the parentheses.
• Next, rearranging terms by moving \( 3y \) to the left side to combine all variable terms. This involves subtracting \( 3y \) from both sides.
• Finally, adjusting constants by subtracting 4 from both sides to isolate the variable term on one side.

Through these transformations, the original equation was rearranged to \( x - 3y = -7 \). Understanding how to transform equations is crucial for solving problems efficiently and understanding different equation forms.