Point-Slope Equation: Line Through (-5, 7) & (4, 13)

Let's dive into finding the equation of a line when we're given two points it passes through. Specifically, we'll focus on using the point-slope form. This form is super handy because it directly incorporates a point on the line and the slope of the line. In our case, the line passes through the points $(-5, 7)$ and $(4, 13)$, and we want to use the point $(-5, 7)$ to write the equation. Get ready to explore the step-by-step process!

Understanding Point-Slope Form

The point-slope form of a linear equation is expressed as:

$y - y_1 = m(x - x_1)$

Where:

• $(x_1, y_1)$ is a known point on the line.
• $m$ is the slope of the line.

The beauty of this form is how directly it relates the geometric properties (a point and slope) to the algebraic equation of the line. You can immediately visualize the line if you know a point and its slope.

Step 1: Calculate the Slope

Before we can use the point-slope form, we need to find the slope ($m$) of the line that passes through the points $(-5, 7)$ and $(4, 13)$. The slope is a measure of the line's steepness and direction. The formula to calculate the slope ($m$) given two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

In our case, $(x_1, y_1) = (-5, 7)$ and $(x_2, y_2) = (4, 13)$. Plugging these values into the formula, we get:

$m = \frac{13 - 7}{4 - (-5)} = \frac{6}{4 + 5} = \frac{6}{9} = \frac{2}{3}$

So, the slope of the line is $\frac{2}{3}$. This means for every 3 units we move to the right on the x-axis, the line rises 2 units on the y-axis. Understanding the slope is crucial for visualizing the line's behavior. The positive slope indicates that the line is increasing.

Step 2: Apply the Point-Slope Form

Now that we have the slope ($m = \frac{2}{3}$) and a point on the line $(-5, 7)$, we can plug these values into the point-slope form of the equation:

$y - y_1 = m(x - x_1)$

Substituting $x_1 = -5$, $y_1 = 7$, and $m = \frac{2}{3}$, we get:

$y - 7 = \frac{2}{3}(x - (-5))$

Simplifying the equation, we have:

$y - 7 = \frac{2}{3}(x + 5)$

This is the equation of the line in point-slope form using the point $(-5, 7)$.

Step 3: Verify the Equation

To ensure our equation is correct, we can plug in the other given point $(4, 13)$ into our point-slope equation and see if it holds true:

$13 - 7 = \frac{2}{3}(4 + 5)$

$6 = \frac{2}{3}(9)$

$6 = 6$

Since the equation holds true, our point-slope equation is correct. Verification is a vital step to ensure accuracy.

Final Answer

The equation of the line in point-slope form, using the point $(-5, 7)$, is:

$y - 7 = \frac{2}{3}(x + 5)$

Thus, the missing values are:

$(y - 7) = \frac{2}{3}(x - (-5))$

This completes the problem. The point-slope form is now fully defined for the given line.

Alternative Forms of a Linear Equation

While the point-slope form is useful, it's beneficial to recognize other forms of linear equations. Each form has its own advantages depending on the information you're given or what you want to emphasize.

Slope-Intercept Form

The slope-intercept form is written as:

$y = mx + b$

Where:

• $m$ is the slope of the line.
• $b$ is the y-intercept (the point where the line crosses the y-axis).

To convert our point-slope form to slope-intercept form, we can distribute and solve for $y$:

$y - 7 = \frac{2}{3}(x + 5)$

$y - 7 = \frac{2}{3}x + \frac{10}{3}$

$y = \frac{2}{3}x + \frac{10}{3} + 7$

$y = \frac{2}{3}x + \frac{10}{3} + \frac{21}{3}$

$y = \frac{2}{3}x + \frac{31}{3}$

So, the slope-intercept form of the equation is $y = \frac{2}{3}x + \frac{31}{3}$. This form directly tells us the slope and y-intercept of the line.

Standard Form

The standard form of a linear equation is written as:

$Ax + By = C$

Where $A$, $B$, and $C$ are constants, and $A$ is usually a positive integer. To convert our slope-intercept form to standard form, we can rearrange the equation:

$y = \frac{2}{3}x + \frac{31}{3}$

Multiply through by 3 to eliminate fractions:

$3y = 2x + 31$

Rearrange to get the standard form:

$-2x + 3y = 31$

Multiply by -1 to make A positive:

$2x - 3y = -31$

So, the standard form of the equation is $2x - 3y = -31$.

Importance of Understanding Linear Equations

Linear equations are fundamental in mathematics and have a wide range of applications in various fields. They are used to model relationships between two variables that exhibit a constant rate of change. Understanding linear equations is crucial for solving problems in algebra, geometry, calculus, and beyond.

Real-World Applications

• Physics: Describing motion with constant velocity.
• Economics: Modeling supply and demand curves.
• Engineering: Designing structures and circuits.
• Computer Science: Creating linear regression models for data analysis.

Tips for Mastering Linear Equations

1. Practice Regularly: The more you practice, the better you'll become at recognizing and solving linear equations.
2. Visualize the Equations: Use graphing tools to visualize the lines and understand their properties.
3. Understand Different Forms: Familiarize yourself with point-slope, slope-intercept, and standard forms.
4. Apply to Real-World Problems: Look for opportunities to apply linear equations to solve real-world problems.

By understanding the different forms of linear equations and practicing regularly, you can master this fundamental concept and apply it to a wide range of problems.

Conclusion

We've successfully found the point-slope form of the equation for a line passing through the points $(-5, 7)$ and $(4, 13)$, using the point $(-5, 7)$. The equation is $y - 7 = \frac{2}{3}(x + 5)$. We also explored the slope-intercept and standard forms of the equation. Understanding these forms and practicing regularly will solidify your understanding of linear equations.

For further learning on linear equations, visit Khan Academy's Linear Equations section.