Problem 49 - Algebra 2

Question

REVIEW What is the equation of the line that has a slope of 3 and passes through the point \((2,-9) ?\) \(\mathbf{F} \quad y=3 x+11\) \(\mathbf{G} y=3 x-11\) \(\mathbf{H} y=3 x+15\) \(\mathbf{J} \quad y=3 x-15\)

Step-by-Step Solution

Answer

The equation is \(y = 3x - 15\) (Choice J).

Step 1: Identify the Given Values

We are given that the slope of the line is 3, and it passes through the point \((2,-9)\).

Step 2: Use the Slope-Intercept Form

The slope-intercept form of a linear equation is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 3: Substitute the Slope

Since the slope \(m\) is 3, we substitute it into the equation to get \(y = 3x + b\).

Step 4: Find the Y-Intercept

Substitute the coordinates of the point \((2, -9)\) into the equation: \[-9 = 3(2) + b\] Simplify to find \(b\).

Step 5: Solve for Y-Intercept

Calculate the terms: \[-9 = 6 + b\] Now, solve for \(b\) by subtracting 6 from both sides:\[-9 - 6 = b\]\[b = -15\]

Step 6: Write the Equation of the Line

With \(b = -15\) and \(m = 3\), the equation of the line is:\[y = 3x - 15\]

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form is essential for writing linear equations. The slope-intercept form is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept, which is the point where the line crosses the y-axis.
This form is highly useful because it directly shows both the slope and the y-intercept. Thus, it provides a clear picture of the line's behavior and position in the coordinate system.

• Slope \(m\): This tells us how steep the line is. A larger absolute value of \(m\) indicates a steeper line.
• Y-intercept \(b\): This tells us where the line crosses the y-axis. It gives a point of reference for drawing the line.

Knowing how to manipulate this form allows you to quickly identify a line's characteristics.

Y-Intercept

The y-intercept is the value of \(y\) where the line crosses the y-axis. In the formula \(y = mx + b\), \(b\) is the y-intercept.
Finding the y-intercept is straightforward if you know a point on the line and the slope.
Take the point \((x_1, y_1)\) and plug it into the equation along with the known slope \(m\):

• The equation becomes \(y_1 = mx_1 + b\).
• Re-arrange the equation to solve for \(b\): \(b = y_1 - mx_1\).

If a line passes through the point \((2, -9)\) and has a slope \(m = 3\), we substitute and solve for \(b\), resulting in \(b = -15\). This means the line crosses the y-axis at \(-15\). This value is crucial to plot and understand the line's position.

Solving Equations

Finding the equation of a line involves solving for the y-intercept using given points and the slope.
Here’s a simple way to approach solving such linear equations:

• Input known values: Start with substituting the slope \(m\) and point \((x_1, y_1)\) into the slope-intercept formula.
• Rearrange terms: Make \(b\) the subject to find the y-intercept.
• Solve for unknowns: Perform arithmetic operations to isolate \(b\).

This process helps in converting point-slope information into a full equation \(y = mx + b\).
Solving equations step by step ensures clarity and precision, allowing you to accurately describe the linear relationship between variables.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses coordinates to represent and solve geometric problems. It bridges algebra with geometry, enabling the visualization of math concepts.
In a coordinate plane, a line's properties are determined by its slope and intercepts. Understanding these helps comprehend how lines are positioned:

• Slope \(m\): Determines the angle and direction of the line.
• X-intercept: The point where the line crosses the x-axis, found by setting \(y = 0\) and solving for \(x\).
• Y-intercept \(b\): As previously mentioned, the point where the line crosses the y-axis.

By applying coordinate geometry, you can easily plot points, lines, and figures, reinforcing how changes in equations affect their graphical representations. This integration promotes a deeper understanding of mathematical relationships and real-world applications.