Problem 25 - Algebra 1: Concepts and Skills

Question

Rewrite the equation in function form. $$ 5 x+5 y=19 $$

Step-by-Step Solution

Answer

The equation rewritten in function form is \(y = 3.8 - x\).

Step 1: Rearrange Equation

Start by subtracting \(5x\) from both sides of the equation to isolate \(5y\). So, the equation becomes \(5y = 19 - 5x\).

Step 2: Isolating y

Next, divide through by 5 to totally isolate 'y'. This gives the equation in function form as \(y = \frac{19 - 5x}{5}\).

Step 3: Simplify Equation

Lastly, the equation can be simplified to \(y = 3.8 - x\).

Key Concepts

Linear Equations

Linear equations are fundamental mathematical expressions that form a straight line when graphed. They are integral in various areas, such as algebra, coordinate geometry, and even real-world problem solving. A linear equation is typically expressed in the general form \( ax + by = c \) where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables.
Linear equations have the power to convey relationships between two variables and show how they change together. For example, in our original equation \( 5x + 5y = 19 \), the coefficients of \( x \) and \( y \) indicate how changes in one variable affect the other.

Key Characteristics of Linear Equations include:

• Graphing: They form a straight line.
• Variables: Typically include two, such as \( x \) and \( y \).
• Solution: Finding the point(s) where lines intersect, either with each other or with axes.

Understanding linear equations allows us to model and solve numerous practical situations, making it a crucial skill in mathematics.

Rearranging Equations

Rearranging equations involves adjusting an equation to isolate a particular variable of interest. This process is essential in both mathematics and many fields like physics and economics, where solving for one variable can simplify analysis or demonstrate specific relationships.
In the example equation \( 5x + 5y = 19 \), we rearranged the terms to solve for \( y \). This step is especially useful when you want to express a variable as a function of others, often seen in function form equations.

Key steps in rearranging equations:

• Identify the target variable: Determine which variable you want to isolate.
• Use inverse operations: Subtract, add, multiply, or divide both sides of the equation to simplify.
• Maintain balance: What you do to one side must be done to the other to keep the equation valid.

Mastering the rearrangement of equations enhances your mathematical problem-solving skills and supports more complex algebraic operations.

Solving for y

Solving for \( y \) means isolating \( y \) on one side of the equation to express it as a function of \( x \). This is a crucial technique in algebra that helps understand relationships and dependencies between variables.
In our example, the equation was \( 5x + 5y = 19 \). By rearranging it and performing operations, we expressed \( y \) in terms of \( x \) as \( y = 3.8 - x \).

Steps in solving for \( y \):

• Move other terms: Subtract the terms involving \( x \) from both sides if necessary.
• Isolate \( y \): Once terms involving \( y \) are isolated, divide or multiply to solve for \( y \).
• Simplify the expression: Make sure the final expression is in its simplest form.

This technique is particularly useful for graphing functions and understanding the slope-intercept form, where \( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept, offering a clear view of how changes in \( x \) affect \( y \).