Problem 70 - Algebra 1: Concepts and Skills

Question

Find the slope and the y-intercept of the graph of the equation. Then graph the equation. $$ 9 x+3 y=15 $$

Step-by-Step Solution

Answer

The slope of the line is -3 and the y-intercept is 5.

Step 1: Put the Equation in Slope-Intercept Form

To do this for the equation \(9x + 3y = 15\), subtract \(9x\) from both sides of the equation. This gives the equation in slope-intercept form of \(y = -3x + 5\).

Step 2: Identify the Slope and Y-Intercept

In the equation \(y = -3x + 5\), '-3' is the coefficient of 'x', so the slope (m) is -3. The y-intercept (c) is the constant term, which is 5.

Step 3: Graph the Equation

Start by plotting the y-intercept point (0,5) on the graph. Since the slope is -3, for every 3 units decrease in y, increase x by 1 unit. Do the reverse for every 3 units increase in y, decrease x by 1 unit. Use these points to sketch the line.

Key Concepts

graphing linear equations

Graphing linear equations is a fundamental skill in algebra that helps us visualize relationships between variables. At its core, a linear equation creates a straight line on a graph. The most common way to express a linear equation is in the slope-intercept form, which is \(y = mx + c\). This form provides a clear way to understand the properties of the line.

• \(m\) represents the slope of the line.
• \(c\) is the y-intercept, the point where the line crosses the y-axis.

To graph a linear equation:

• Start by locating the y-intercept on the y-axis.
• Use the slope to determine the direction and steepness of the line.
• Draw the line through the points to create a complete graph.

Breaking the steps into manageable tasks can simplify the graphing process, especially when handling more complex equations.

slope

The slope of a line, denoted as \(m\) in the slope-intercept form, is a measure of its steepness. It describes how much the line rises or falls as it moves from left to right.

• A positive slope means the line ascends as it moves from left to right.
• A negative slope means the line descends as it moves from left to right.
• The greater the absolute value of the slope, the steeper the line.
• If the slope is zero, the line is horizontal, indicating no change in y as x changes.

In the example equation \(y = -3x + 5\), the slope is \(-3\). This means for every unit increase in x, the value of y decreases by 3 units. Understanding the slope is crucial for predicting how the line behaves as it stretches across the coordinate plane.

y-intercept

The y-intercept is a fundamental part of graphing linear equations. It is the point where the line crosses the y-axis, which occurs when \(x=0\). In the slope-intercept form \(y = mx + c\), \(c\) represents the y-intercept. Identifying the y-intercept ishelpful because:

• It provides a starting point for drawing the line on a graph.
• It allows us to interpret the location of the line relative to the coordinate system.

In the given equation \(y = -3x + 5\), the y-intercept is \(5\). This means the line crosses the y-axis at the point (0, 5). By plotting this point first and using the slope to find other points, we can accurately sketch the line on a graph. The y-intercept gives us a practical and visual foothold in determining the line's alignment on the graph.