Problem 61 - Calculus Single Variable

Question

Find the point at which the lines determined by the two given equations intersect. $$ -2 x+5 y=38,5 x+2 y=-8 $$

Step-by-Step Solution

Answer

Solve the equations for an intersection point.

Step 1: Understand the System of Equations

The problem involves two equations: - Equation 1: \(-2x + 5y = 38\) - Equation 2: \(5x + 2y = -8\).These represent two lines, and we need to find their intersection point. This point will satisfy both equations simultaneously.

Step 2: Solve One Equation for One Variable

First, solve Equation 1 for one of the variables. Let's solve for \(x\):\[-2x + 5y = 38\]Add \(2x\) to both sides:\[5y = 2x + 38\]Then isolate \(x\):\[2x = 5y - 38\]Divide by 2:\[x = \frac{5y - 38}{2}\].

Key Concepts

Intersection of Lines

In mathematics, the point where two lines meet is known as the intersection. This is a crucial concept when dealing with systems of linear equations. When graphs of two linear equations cross each other, they intersect at a specific point that satisfies both equations.
Understanding this point as a solution, it represents the only pair of coordinates that both lines share. To find this point, it is necessary to solve both equations simultaneously. This involves using algebraic methods, such as substitution or elimination, to determine values for both variables (typically, \(x\) and \(y\)).
Once these values are determined, they can be plugged back into the original equations to verify the solution. If the values satisfy both equations, you have found the precise coordinates of the intersection.

Linear Equations

Linear equations are fundamental components of algebra. These equations describe straight lines when graphed on a coordinate plane. Each linear equation can typically be written in the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants.
Every linear equation consists of two variables, usually \(x\) and \(y\). The coefficients of these variables determine the slope and position of the line on the graph. Slope-intercept form used frequently in these problems corresponds to \(y = mx + b\), where \(m\) represents the slope, and \(b\) represents the y-intercept.
Linear equations are not just theoretical constructs; they are practical tools used in many fields to model real-world situations, like calculating interest, predicting expenses, or analyzing trends.

Algebraic Methods

Algebraic methods are strategies that enable solving systems of equations. Two common approaches are substitution and elimination, both being effective depending on the context of the problem.

• Substitution: Involves solving one equation for a particular variable and then substituting that expression into another equation. This reduces the system to a single equation with one variable, making it simpler to solve.


• Elimination: A method where you add or subtract equations to eliminate a variable. This often involves manipulating the equations by multiplying them by constants so that their coefficients align, allowing one variable to cancel out.

Both methods aim at finding the values of the variables that fulfill both equations. The choice between them often depends on the specific structure of the equations involved in the problem.