Finding Equations Of Lines Intersection And Slope

Jul 11, 2025 by Admin 50 views

Finding the Equation of a Line with Intersection and Slope Conditions

In mathematics, determining the equations of lines based on given conditions is a fundamental skill. This article delves into a specific problem where we need to find the equation of a line given its slope and the $y$ -coordinate of its intersection point with another line. This type of problem combines concepts from linear equations, coordinate geometry, and algebraic manipulation, making it a valuable exercise for understanding the relationships between different mathematical ideas.

Problem Statement

The point where the graphs of two equations intersect has a $y$ -coordinate of 2. One equation is given as $y = -3x + 5$ . We are tasked with finding the other equation, knowing that its graph has a slope of 1. This problem requires us to use the given information to first find the $x$ -coordinate of the intersection point and then use the slope and point to determine the equation of the second line.

Step-by-Step Solution

To solve this problem effectively, we will follow a step-by-step approach. First, we need to identify the coordinates of the intersection point by utilizing the given $y$ -coordinate and the equation of the first line. Once we have the coordinates of the intersection point, we can use this point along with the given slope of the second line to determine its equation. This involves using the point-slope form of a linear equation, which is a crucial concept in coordinate geometry. The steps are as follows:

Find the x-coordinate of the intersection point: We know that the $y$ -coordinate of the intersection point is 2. We can substitute this value into the equation of the first line, $y = -3x + 5$ , to find the corresponding $x$ -coordinate. This involves solving a simple linear equation for $x$ .
Determine the coordinates of the intersection point: Once we have both the $x$ and $y$ coordinates, we can express the intersection point as an ordered pair $(x, y)$ . This point lies on both lines, which is a fundamental concept in understanding simultaneous equations.
Use the point-slope form to find the second equation: We know the slope of the second line is 1, and we now have a point that lies on this line. We can use the point-slope form of a linear equation, which is given by $y - y_1 = m(x - x_1)$ , where $m$ is the slope and $(x_1, y_1)$ is a point on the line. Substituting the known values will give us the equation of the second line.
Simplify the equation: After substituting the values into the point-slope form, we need to simplify the equation to the slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the $y$ -intercept. This simplification involves algebraic manipulation to isolate $y$ on one side of the equation.

Detailed Solution

Let's begin by finding the $x$ -coordinate of the intersection point. We substitute $y = 2$ into the equation $y = -3x + 5$ :

$2 = -3x + 5$

To solve for $x$ , we first subtract 5 from both sides:

$2 - 5 = -3x$

$-3 = -3x$

Now, we divide both sides by -3:

$x = 1$

So, the $x$ -coordinate of the intersection point is 1. Therefore, the intersection point is $(1, 2)$ .

Next, we use the point-slope form of a linear equation to find the equation of the second line. The point-slope form is given by:

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

where $m$ is the slope and $(x_1, y_1)$ is a point on the line. We know the slope $m = 1$ and the point $(1, 2)$ , so we substitute these values into the point-slope form:

$y - 2 = 1(x - 1)$

Now, we simplify the equation to the slope-intercept form, $y = mx + b$ . First, distribute the 1 on the right side:

$y - 2 = x - 1$

Next, add 2 to both sides:

$y = x - 1 + 2$

$y = x + 1$

Thus, the equation of the second line is $y = x + 1$ .

Verification

To verify our solution, we need to ensure that the point $(1, 2)$ lies on both lines. We already know that it lies on the first line because we used the $y$ -coordinate of 2 to find the $x$ -coordinate. Let's check if it lies on the second line, $y = x + 1$ :

Substitute $x = 1$ into the equation:

$y = 1 + 1$

$y = 2$

Since the $y$ -coordinate is 2, the point $(1, 2)$ does indeed lie on the second line. Therefore, our solution is correct.

Alternative Approaches

While we have used the point-slope form to find the equation of the second line, there are alternative approaches we could have taken. One such approach is to directly use the slope-intercept form, $y = mx + b$ . We know the slope $m = 1$ , so the equation becomes:

$y = 1x + b$

$y = x + b$

We also know that the point $(1, 2)$ lies on this line, so we can substitute the coordinates into the equation to solve for $b$ :

$2 = 1 + b$

Subtract 1 from both sides:

$b = 1$

So, the equation of the line is $y = x + 1$ , which matches our previous result. This alternative method provides a slightly different perspective on the problem and can be useful in other similar situations.

Common Mistakes

When solving problems of this type, there are several common mistakes that students often make. These include:

Incorrectly substituting values: It is crucial to substitute the values correctly when finding the $x$ -coordinate or using the point-slope form. A common mistake is to mix up the $x$ and $y$ values or to substitute them into the wrong equation.
Algebraic errors: Errors in algebraic manipulation can lead to incorrect results. This includes mistakes in solving for $x$ , distributing terms, or simplifying equations. It is important to double-check each step to avoid these errors.
Misunderstanding the point-slope form: The point-slope form is a powerful tool, but it needs to be used correctly. A common mistake is to incorrectly identify the slope or the point on the line.
Not simplifying the equation: After using the point-slope form, it is essential to simplify the equation to the slope-intercept form. Failing to do so can lead to a final answer that is not in the most useful form.
Forgetting to verify the solution: Always verify the solution by checking that the point of intersection lies on both lines. This helps to catch any errors that may have been made during the solution process.

Importance of Understanding Linear Equations

Understanding linear equations is crucial in mathematics and has numerous real-world applications. Linear equations are used to model relationships between two variables that change at a constant rate. They are fundamental in fields such as physics, engineering, economics, and computer science.

Modeling real-world situations: Linear equations can be used to model a wide range of real-world situations, such as the relationship between time and distance, the cost of goods and the quantity purchased, or the relationship between temperature and pressure. Understanding linear equations allows us to make predictions and analyze these situations more effectively.
Solving practical problems: Many practical problems can be solved using linear equations. For example, determining the optimal price for a product, calculating the trajectory of a projectile, or designing a structure that can withstand certain forces.
Foundation for advanced mathematics: Linear equations serve as a foundation for more advanced mathematical concepts, such as systems of equations, matrices, and linear algebra. A strong understanding of linear equations is essential for success in these areas.
Graphical representation: The graphical representation of linear equations provides a visual way to understand the relationship between the variables. The slope and intercepts of the line give valuable information about the relationship, such as the rate of change and the starting value.

Practice Problems

To reinforce the concepts discussed in this article, it is helpful to practice similar problems. Here are a few practice problems:

The point where the graphs of two equations intersect has an $x$ -coordinate of 3. One equation is $y = 2x - 1$ . Find the other equation if its graph has a slope of -2.
The point where the graphs of two equations intersect has a $y$ -coordinate of -1. One equation is $y = -x + 2$ . Find the other equation if its graph has a slope of 3.
The point where the graphs of two equations intersect is $(2, 5)$ . One equation has a slope of 1 and the other has a slope of -1. Find the equations of both lines.

Solving these problems will help solidify your understanding of linear equations and the techniques used to find the equation of a line given certain conditions.

Conclusion

Finding the equation of a line given the $y$ -coordinate of its intersection point with another line and its slope is a valuable mathematical exercise. It requires a solid understanding of linear equations, coordinate geometry, and algebraic manipulation. By following a step-by-step approach, substituting values correctly, and simplifying the equation, we can successfully solve this type of problem. Understanding linear equations is fundamental in mathematics and has numerous real-world applications, making it an essential skill for students to develop. Remember to practice similar problems to reinforce your understanding and to avoid common mistakes. By mastering these concepts, you will be well-prepared to tackle more complex mathematical challenges.