Finding The Intersection Point Of Linear Equations A Step-by-Step Guide

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In mathematics, particularly in algebra, finding the point of intersection between two or more linear equations is a fundamental concept. This point represents the solution that satisfies all the equations simultaneously. Understanding how to find these intersection points is crucial for solving various real-world problems, from determining break-even points in business to predicting traffic flow in urban planning. This article provides a detailed guide on how to find the point of intersection for a pair of linear equations, focusing on the specific example provided:

\begin{cases}
  x + y = -7.8 \\
  y = 2x + 8.1
\end{cases}

We will explore different methods to solve this system of equations, including substitution and elimination, and discuss the underlying principles that make these methods effective. By the end of this article, you will have a clear understanding of how to solve similar problems and interpret the results.

Before diving into the methods for finding the intersection point, it's essential to understand what linear equations are and how they represent lines on a graph. Linear equations are algebraic expressions that, when graphed on a coordinate plane, produce a straight line. The general form of a linear equation is y = mx + b, where m represents the slope of the line and b represents the y-intercept (the point where the line crosses the y-axis). Alternatively, linear equations can also be expressed in the standard form Ax + By = C, where A, B, and C are constants. Each form provides valuable information about the line's characteristics and its position on the coordinate plane.

In the given problem, we have two linear equations:

  1. x + y = -7.8
  2. y = 2x + 8.1

The first equation, x + y = -7.8, is in the standard form, while the second equation, y = 2x + 8.1, is in the slope-intercept form. Each equation represents a unique line on the coordinate plane. The point where these two lines intersect is the solution that satisfies both equations simultaneously. This means that the x and y coordinates of the intersection point will make both equations true.

The graphical interpretation of the intersection point is straightforward: it is the single point where the two lines cross each other. However, finding this point graphically can be imprecise, especially when the coordinates are not integers. Therefore, algebraic methods are preferred for their accuracy and efficiency. In the following sections, we will delve into the most common algebraic methods for solving systems of linear equations: the substitution method and the elimination method. These methods provide a systematic approach to finding the exact coordinates of the intersection point, ensuring a precise solution to the problem.

There are primarily two algebraic methods to find the point of intersection for a pair of linear equations: the substitution method and the elimination method. Both methods aim to solve for the variables x and y that satisfy both equations. Let's explore each method in detail.

Substitution Method

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system of two equations with two variables into a single equation with one variable, which can then be easily solved. Once the value of one variable is found, it can be substituted back into either of the original equations to find the value of the other variable. The substitution method is particularly effective when one of the equations is already solved for one variable, as is the case in our example.

For the given system of equations:

\begin{cases}
  x + y = -7.8 \\
  y = 2x + 8.1
\end{cases}

The second equation, y = 2x + 8.1, is already solved for y. This makes the substitution method a natural choice. We can substitute the expression for y from the second equation into the first equation:

x + (2x + 8.1) = -7.8

Now, we have a single equation with one variable, x. Simplify and solve for x:

3x + 8.1 = -7.8

3x = -7.8 - 8.1

3x = -15.9

x = -15.9 / 3

x = -5.3

Now that we have the value of x, we can substitute it back into either of the original equations to find the value of y. Using the second equation, y = 2x + 8.1:

y = 2(-5.3) + 8.1

y = -10.6 + 8.1

y = -2.5

Therefore, the point of intersection, using the substitution method, is (-5.3, -2.5). This means that the values x = -5.3 and y = -2.5 satisfy both equations simultaneously.

Elimination Method

The elimination method, also known as the addition method, involves manipulating the equations so that when they are added together, one of the variables is eliminated. This is achieved by multiplying one or both equations by a constant so that the coefficients of one of the variables are opposites. When the equations are added, the variable with opposite coefficients cancels out, leaving a single equation with one variable. This equation can then be solved, and the value can be substituted back into one of the original equations to find the value of the other variable.

To apply the elimination method to the given system of equations:

\begin{cases}
  x + y = -7.8 \\
  y = 2x + 8.1
\end{cases}

First, we need to rewrite the second equation in the standard form Ax + By = C. Subtracting 2x from both sides of the second equation gives:

-2x + y = 8.1

Now, we have the system:

\begin{cases}
  x + y = -7.8 \\
  -2x + y = 8.1
\end{cases}

To eliminate y, we can multiply the first equation by -1:

-1(x + y) = -1(-7.8)

-x - y = 7.8

Now, we have the system:

\begin{cases}
  -x - y = 7.8 \\
  -2x + y = 8.1
\end{cases}

Adding the two equations together eliminates y:

(-x - y) + (-2x + y) = 7.8 + 8.1

-3x = 15.9

x = -5.3

Now that we have the value of x, we can substitute it back into either of the original equations to find the value of y. Using the first equation, x + y = -7.8:

-5.3 + y = -7.8

y = -7.8 + 5.3

y = -2.5

Therefore, the point of intersection, using the elimination method, is (-5.3, -2.5). This confirms the result obtained using the substitution method.

In this section, we will explicitly demonstrate the steps to solve the given system of equations using both the substitution and elimination methods. This will provide a clear and step-by-step guide for readers to follow and understand the application of these methods.

Step-by-Step Solution Using Substitution

Given the system of equations:

\begin{cases}
  x + y = -7.8 \\
  y = 2x + 8.1
\end{cases}

  1. Identify the equation that is already solved for one variable: The second equation, y = 2x + 8.1, is already solved for y.

  2. Substitute the expression for the variable into the other equation: Substitute 2x + 8.1 for y in the first equation: x + (2x + 8.1) = -7.8

  3. Simplify and solve for the remaining variable: Combine like terms: 3x + 8.1 = -7.8 Subtract 8.1 from both sides: 3x = -15.9 Divide by 3: x = -5.3

  4. Substitute the value found back into one of the original equations to solve for the other variable: Substitute x = -5.3 into the second equation, y = 2x + 8.1: y = 2(-5.3) + 8.1 y = -10.6 + 8.1 y = -2.5

  5. Write the solution as an ordered pair (x, y): The point of intersection is (-5.3, -2.5).

Step-by-Step Solution Using Elimination

Given the system of equations:

\begin{cases}
  x + y = -7.8 \\
  y = 2x + 8.1
\end{cases}

  1. Rewrite the equations in standard form (Ax + By = C): The first equation is already in standard form: x + y = -7.8 Rewrite the second equation by subtracting 2x from both sides: -2x + y = 8.1

  2. Multiply one or both equations by a constant so that the coefficients of one variable are opposites: To eliminate y, multiply the first equation by -1: -1(x + y) = -1(-7.8) -x - y = 7.8

  3. Add the equations together to eliminate one variable: Add the modified first equation to the second equation: (-x - y) + (-2x + y) = 7.8 + 8.1 -3x = 15.9

  4. Solve for the remaining variable: Divide by -3: x = -5.3

  5. Substitute the value found back into one of the original equations to solve for the other variable: Substitute x = -5.3 into the first equation, x + y = -7.8: -5.3 + y = -7.8 Add 5.3 to both sides: y = -2.5

  6. Write the solution as an ordered pair (x, y): The point of intersection is (-5.3, -2.5).

After finding the point of intersection, it is crucial to verify the solution to ensure its accuracy. This can be done by substituting the x and y coordinates back into both original equations. If the solution is correct, it should satisfy both equations, making them true statements. This verification step is essential to avoid errors and confirm the correctness of the solution.

For the given system of equations and the solution (-5.3, -2.5), we will substitute x = -5.3 and y = -2.5 into each equation:

  1. x + y = -7.8 (-5.3) + (-2.5) = -7.8 -7.8 = -7.8 (True)

  2. y = 2x + 8.1 (-2.5) = 2(-5.3) + 8.1 -2.5 = -10.6 + 8.1 -2.5 = -2.5 (True)

Since the solution (-5.3, -2.5) satisfies both equations, it is indeed the correct point of intersection. This verification process provides confidence in the accuracy of the solution and ensures that no algebraic errors were made during the solving process. In practice, verifying the solution is a highly recommended step for any system of equations, especially in situations where accuracy is paramount.

Visualizing the solution graphically can provide a deeper understanding of the concept of intersection points. Each linear equation represents a straight line on the coordinate plane, and the point of intersection is the location where these lines cross each other. Graphing the equations can help confirm the algebraic solution and provide a visual representation of the system.

To graph the given equations:

  1. x + y = -7.8
  2. y = 2x + 8.1

We can plot the lines on a coordinate plane. For the first equation, we can rewrite it in slope-intercept form (y = mx + b) as y = -x - 7.8. This line has a slope of -1 and a y-intercept of -7.8. We can plot the y-intercept and use the slope to find other points on the line.

For the second equation, y = 2x + 8.1, the slope is 2, and the y-intercept is 8.1. Again, we can plot the y-intercept and use the slope to find additional points on the line.

When we plot both lines on the same coordinate plane, we will observe that they intersect at the point (-5.3, -2.5). This graphical representation visually confirms the algebraic solution we found earlier using both the substitution and elimination methods. The intersection point represents the unique solution that satisfies both equations simultaneously.

Graphing can also help identify cases where there are no solutions (parallel lines) or infinitely many solutions (the same line). In the case of parallel lines, the equations will have the same slope but different y-intercepts, resulting in no intersection point. If the equations represent the same line, they will overlap completely, indicating infinitely many solutions.

In summary, the graphical representation of linear equations provides a valuable tool for understanding and verifying solutions to systems of equations. It complements the algebraic methods and offers a visual confirmation of the intersection point.

Finding the point of intersection of linear equations is not just a theoretical exercise; it has numerous practical applications in various fields. Understanding how to solve systems of equations can help in decision-making and problem-solving in real-world scenarios. Let's explore some key applications:

  1. Business and Economics:

    • Break-Even Analysis: In business, the break-even point is where total revenue equals total costs. This can be modeled using linear equations for cost and revenue. The point of intersection represents the break-even point, which is crucial for determining the sales volume needed to cover costs.
    • Supply and Demand: Economic models often use linear equations to represent supply and demand curves. The equilibrium point, where supply equals demand, is the point of intersection of these curves. This point determines the market price and quantity of a product.

  2. Science and Engineering:

    • Physics: In physics, the motion of objects can be described using linear equations. For example, the meeting point of two objects moving along straight paths can be found by solving a system of linear equations representing their positions over time.
    • Engineering: Linear equations are used in structural analysis, circuit analysis, and other engineering fields. Finding the intersection points can help determine stresses in structures, currents in circuits, and other critical parameters.

  3. Everyday Life:

    • Budgeting: Linear equations can help in budgeting and financial planning. For example, comparing the costs of two different phone plans can be done by setting up linear equations for the total cost as a function of usage and finding the point at which one plan becomes more economical than the other.
    • Travel Planning: Determining the meeting point of two travelers moving at different speeds and starting from different locations can be solved using linear equations.

These examples highlight the versatility and importance of understanding how to find the point of intersection of linear equations. Whether it's making informed business decisions, solving scientific problems, or planning everyday activities, this fundamental concept provides a powerful tool for analysis and problem-solving.

In conclusion, finding the point of intersection for a pair of linear equations is a fundamental concept in mathematics with wide-ranging applications. This article has provided a comprehensive guide on how to solve such problems, focusing on the specific example:

\begin{cases}
  x + y = -7.8 \\
  y = 2x + 8.1
\end{cases}

We have explored two primary algebraic methods: the substitution method and the elimination method. Both methods yield the same solution, (-5.3, -2.5), which represents the point where the two lines intersect on the coordinate plane. The step-by-step solutions demonstrated the application of each method, providing a clear and practical guide for readers to follow. Additionally, we emphasized the importance of verifying the solution by substituting the coordinates back into the original equations to ensure accuracy.

The graphical representation of the linear equations further reinforced the concept, allowing for a visual confirmation of the algebraic solution. By plotting the lines on a coordinate plane, we can see that they indeed intersect at the point (-5.3, -2.5).

Moreover, we discussed the real-world applications of finding intersection points in various fields such as business, economics, science, engineering, and everyday life. These examples highlighted the practical significance of this mathematical concept in solving real-world problems and making informed decisions.

Understanding how to find the point of intersection of linear equations is not just an academic exercise; it is a valuable skill that can be applied in numerous contexts. By mastering the methods and concepts discussed in this article, readers will be well-equipped to tackle similar problems and appreciate the power of linear equations in modeling and solving real-world scenarios. Whether you are a student learning algebra, a professional in a technical field, or simply someone looking to enhance your problem-solving skills, the ability to find intersection points is a valuable asset.

The correct answer is B. (-5.3, -2.5).