Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, express the equations in slope intercept form (rounding each number to the thousandths place). Enter this into a graphing calculator as \(Y 1,\) then adjust the ymin and ymax values for your window to include where the y-intercept occurs. State your ymin and ymax values. $$ 4,500 x-200 y=9,528 $$

Step-by-Step Solution

Answer

Equation: \( y = 22.5x - 47.64 \). Set ymin = -50 and ymax = 10.

Step 1: Rearrange the Equation into Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Start by arranging the given equation, \( 4500x - 200y = 9528 \), to solve for \( y \). First, subtract \( 4500x \) from both sides: \(-200y = -4500x + 9528\).

Step 2: Solve for y

Now, divide every term by \(-200\) to solve for \( y \): \( y = \frac{4500}{200}x - \frac{9528}{200} \). Simplifying these fractions gives \( y = 22.5x - 47.64 \).

Step 3: Input Equation into Graphing Calculator

Enter the equation \( y = 22.5x - 47.64 \) into your graphing calculator as \( Y_1 \). This is your equation in slope-intercept form.

Step 4: Determine Y-Intercept for Graphing

The y-intercept is \(-47.64\), so you should ensure your calculator can display this point on the graph.

Step 5: Set Ymin and Ymax

Adjust the graphing calculator settings to include the y-intercept. Set your \( ymin \) to a value a bit lower than \(-47.64\), such as \(-50\). Choose \( ymax \) to accommodate the rest of the line; a covering range could be \( 10 \).

Key Concepts

Graphing Equations

Graphing equations is a visual way to understand the relationships between variables. It involves plotting points on a graph that show the relationship defined by an equation. By graphing an equation, you can see how changing one variable affects another. For linear equations, which form a straight line, understanding graphing allows you to visually perceive the slope and y-intercept.

• Start by identifying the equation you need to graph, which in step 2 of our solution is given as \( y = 22.5x - 47.64 \).
• Using a graphing calculator or plotting manually, you plot the line based on the slope and y-intercept from the equation.
• The plotted line will reveal how \( x \) and \( y \) are related through the slope-intercept form equation.

Understanding this process is critical for visual learning and application of algebraic solutions.

Linear Equations

Linear equations represent one of the simplest forms of mathematical expressions. They consist of two variables without any exponents or powers, forming a straight line when graphed. The general form of a linear equation is often given as \( ax + by = c \), which can be rearranged to the slope-intercept form.

• The exercise we worked on starts with the equation \( 4500x - 200y = 9528 \).
• By rearranging and simplifying, we obtained \( y = 22.5x - 47.64 \).
• The equation showcases a linear relationship between \( x \) and \( y \).

Linear equations are foundational in algebra and help describe direct proportional relationships in various real-world situations.

Y-Intercept

The y-intercept is a key feature of the slope-intercept form of a linear equation. It is the point where the line crosses the y-axis, meaning the value of \( y \) when \( x = 0 \). This gives us valuable information on the starting point in many practical applications.

• In the equation \( y = 22.5x - 47.64 \), the y-intercept is \(-47.64\).
• When graphing, you need to ensure the y-intercept is displayed within your chosen graph window.
• Setting values such as \( ymin = -50 \) and \( ymax = 10 \) accommodates this y-intercept well on a graphing calculator.

Recognizing and plotting the y-intercept correctly is vital as it affects the entire graph and interpretation of the equation. Without it, essential characteristics of the linear equation could be misrepresented or misunderstood.