Problem 42
Question
Find the point where the lines intersect. $$l_{1}: 5 x-6 y+1=0, \quad l_{2}: 8 x+5 y+2=0$$
Step-by-Step Solution
Verified Answer
The point at which the given lines \(l_{1}\) and \(l_{2}\) intersect is at (-\frac{17}{73}, -\frac{2}{73})
1Step 1: Understanding the Problem and Formulating Equations
We are given two equations of lines \(l_{1}\): 5x-6y+1=0 and \(l_{2}\): 8x+5y+2=0 and we are to find the point where both these lines intersect. Express these equations in terms of \(x\) : \n\(l_{1}\) => \(5x = 6y - 1 => x = \frac{(6y - 1)}{5}\) and \(l_{2}\) => \(8x = -5y -2 => x = \frac{(-5y - 2)}{8}\)
2Step 2: Setting Equations Equal to Each Other
Since both equations are equal to \(x\), therefore we can equate them. \nHence, \(\frac{(6y - 1)}{5} = \frac{(-5y - 2)}{8}\)
3Step 3: Solving for Y
Cross multiply to eliminate denominators and then rearrange the equation to solve for \(y\). \nPerforming operations, we get: \n\(8*(6y - 1) = 5*(-5y - 2) => 48y - 8 = -25y - 10 => 73y = -2 => y = -\frac{2}{73}\)
4Step 4: Substituting Y Value into Line Equations
Substitute \(y = -\frac{2}{73}\) into \(l_{1}\) and \(l_{2}\) equations to solve for \(x\). Doing so with \(l_{1}\) yields: \n\(5x = 6*(-\frac{2}{73}) -1 => 5x = -\frac{12}{73} -1 => 5x = - \frac{85}{73} => x = -\frac{17}{73}\). So, we got the same \(x\) value with either line equations.
5Step 5: Cross-Checking
Cross check this point (-\frac{17}{73}, -\frac{2}{73}) with both line equations. If the point satisfies both the equations, then it is the intersection point of both lines.
Key Concepts
System of Linear EquationsSolving Linear EquationsCoordinate Geometry
System of Linear Equations
Understanding the system of linear equations is crucial when dealing with algebra and coordinate geometry. A system of linear equations consists of two or more equations set equal to each other and can be visualized as lines on a coordinate plane. To find the solution to this system, you're looking for a point or points where these lines intersect, which represents the values that satisfy all the equations in the system simultaneously.
For instance, in our problem, we are dealing with two equations, each representing a line on the plane. The equations given by
\(l_{1}: 5x - 6y + 1 = 0\) and \(l_{2}: 8x + 5y + 2 = 0\), form a system that we solve to find their point of intersection. When solving a system of linear equations, it's important to arrange the equations consistently and clearly to avoid confusion, ensuring that the corresponding variables align vertically when written down.
The intersection point provides the values of \(x\) and \(y\) that satisfy both equations, which is the key concept to recognize when approaching these kinds of problems.
For instance, in our problem, we are dealing with two equations, each representing a line on the plane. The equations given by
\(l_{1}: 5x - 6y + 1 = 0\) and \(l_{2}: 8x + 5y + 2 = 0\), form a system that we solve to find their point of intersection. When solving a system of linear equations, it's important to arrange the equations consistently and clearly to avoid confusion, ensuring that the corresponding variables align vertically when written down.
The intersection point provides the values of \(x\) and \(y\) that satisfy both equations, which is the key concept to recognize when approaching these kinds of problems.
Solving Linear Equations
Solving linear equations involves finding the values of variables that make the equation true. When dealing with a single linear equation, we isolate the variable on one side of the equation to identify its value. However, when working with two or more equations, we apply methods such as substitution, elimination, graphing, or matrix operations to solve the system.
The step-by-step solution provided demonstrates the substitution method. After rearranging each equation for \(x\) in terms of \(y\), we set them equal to each other. This is because at the intersection, both \(x\) and \(y\) must have the same value in both equations. The process continues by cross-multiplying to eliminate fractions, which simplifies the task of solving for \(y\).
Once \(y\) is found, you substitute it back into either original equation to solve for \(x\), giving you the coordinated intersection point. Throughout each step, keeping the work neat and checking each operation reduces the risk of errors—a critical advice for successfully solving linear equations.
The step-by-step solution provided demonstrates the substitution method. After rearranging each equation for \(x\) in terms of \(y\), we set them equal to each other. This is because at the intersection, both \(x\) and \(y\) must have the same value in both equations. The process continues by cross-multiplying to eliminate fractions, which simplifies the task of solving for \(y\).
Once \(y\) is found, you substitute it back into either original equation to solve for \(x\), giving you the coordinated intersection point. Throughout each step, keeping the work neat and checking each operation reduces the risk of errors—a critical advice for successfully solving linear equations.
Coordinate Geometry
Coordinate geometry is the branch of mathematics that combines algebra and geometry to discuss points, lines, and shapes using a coordinate system. The fundamental element of coordinate geometry is the coordinate plane, consisting of a horizontal axis (the x-axis) and a vertical axis (the y-axis). The intersection of these axes is the origin, designated as the point (0,0).
In the context of our exercise, we use coordinate geometry principles to graph the lines represented by the linear equations and find their intersection point. Each equation corresponds to a straight line on this plane, and their characteristics—such as slope and y-intercept—can be used for graphing. The coordinates of any point on the line must satisfy the equation representing that line.
When the equations of two lines are given, finding the intersection point algebraically, as shown in the solution steps, is equivalent to finding the point where the lines would cross if graphed. It's essential to understand that the intersection point's coordinates are the solution to the system of equations, providing a powerful connection between algebraic methods and geometric interpretation.
In the context of our exercise, we use coordinate geometry principles to graph the lines represented by the linear equations and find their intersection point. Each equation corresponds to a straight line on this plane, and their characteristics—such as slope and y-intercept—can be used for graphing. The coordinates of any point on the line must satisfy the equation representing that line.
When the equations of two lines are given, finding the intersection point algebraically, as shown in the solution steps, is equivalent to finding the point where the lines would cross if graphed. It's essential to understand that the intersection point's coordinates are the solution to the system of equations, providing a powerful connection between algebraic methods and geometric interpretation.
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