Problem 45 - Algebra 1: Concepts and Skills

Question

Find the x-intercepts and the y-intercepts of the line. Graph the equation. Label the points where the line crosses the axes. $$ 36 x+4 y=44 $$

Step-by-Step Solution

Answer

The x-intercept is \( \frac{11}{9}, 0 \) and the y-intercept is \(0, 11\). The line crosses the axes at these points.

Step 1: Solve for the X-intercept

Set y to zero in the equation and solve for x: \[36x + 4(0) = 44 \rightarrow x = \frac{44}{36} = \frac{11}{9}\]. So the x-intercept is \( \frac{11}{9}, 0 \).

Step 2: Solve for the Y-intercept

Now, set x to zero in the equation and solve for y: \[36(0) + 4y = 44 \rightarrow y = \frac{44}{4} = 11\]. So the y-intercept is \(0, 11\).

Step 3: Graph the Equation

After obtaining the x and y intercepts, draw these points on graph paper, then draw a line that passes through these points. Label the points \( \frac{11}{9}, 0 \) and \(0, 11\), which are the x-intercept and y-intercept respectively.

Key Concepts

Intercepts

Intercepts are the points where a line crosses the x-axis and y-axis on a graph. Identifying these points is a crucial step in graphing a linear equation.

• The x-intercept is found by setting the y value to zero in the equation and solving for x. In the equation \( 36x + 4y = 44 \), we set \( y = 0 \), which simplifies to \( 36x = 44 \). Solving for \( x \) gives us the point \( \left( \frac{11}{9}, 0 \right) \).
• The y-intercept is found by setting the x value to zero and solving for y, which gives us \( 4y = 44 \). This simplifies to \( y = 11 \), giving us the point \( (0, 11) \).

These intercepts provide concise information about where the line meets the axes, making them essential to plot accurately on a graph.

Graphing

Graphing a linear equation involves visually representing the equation on a coordinate plane. Here is how you can achieve this step by step.

• Start by plotting the intercepts you've calculated: place a point at the x-intercept \( \left( \frac{11}{9}, 0 \right) \) and another at the y-intercept \( (0, 11) \).
• Once the points are on the graph, draw a straight line through them. This line represents all the solutions to the equation \( 36x + 4y = 44 \).

The clear advantage of graphing is that it provides a visual understanding of the relationship between x and y. Knowing this, you can readily identify trends such as the positive or negative direction of the line, the steepness or slope, and where it crosses the axes.

Equation of a Line

The equation of a line is a formula that describes the entirety of the line on a graph. The general form of a linear equation, such as \( 36x + 4y = 44 \), is known as the standard form.

• Standard form equations are useful because they clearly show both x and y coefficients, aiding in finding intercepts directly.
• If required, converting from standard form to slope-intercept form \( y = mx + b \) provides the slope \( m \) and the y-intercept \( b \), which can offer more direct insight into the line's behavior.

Understanding the equation allows us to swiftly calculate specific points on the line and also predict the line's direction and steepness, paving the way for deeper analytical exploration of relationships between variables.