Problem 62 - Calculus Single Variable

Question

Find the point at which the lines determined by the two given equations intersect. $$ y=3 x+1 / 2, y=2 x+1 $$

Step-by-Step Solution

Answer

The intersection point is \( (\frac{1}{2}, 2) \).

Step 1: Set the Equations Equal

To find the intersection of the two lines, the first step is to set their equations equal. Since both equations are expressed as functions of the same variable, we equate the two equations: \( 3x + \frac{1}{2} = 2x + 1 \)

Step 2: Solve for x

We solve the equation from Step 1 for \( x \). First, isolate \( x \) by moving terms involving \( x \) to one side and constant terms to the other:\[3x - 2x = 1 - \frac{1}{2}\]Simplify both sides:\[x = \frac{1}{2}\].

Step 3: Substitute x back into one equation

Substitute \( x = \frac{1}{2} \) back into either of the original equations to find \( y \). Let's use \( y = 2x + 1 \):\[y = 2\left(\frac{1}{2}\right) + 1 = 1 + 1 = 2\]

Step 4: State the Intersection Point

Now that we have both \( x \) and \( y \), state the point of intersection. From the calculations, we find that the lines intersect at the point \( \left( \frac{1}{2}, 2 \right) \).

Key Concepts

Systems of Equations

A system of equations is a set of two or more equations that share two or more unknowns. In this context, we are dealing with linear equations, which graph as straight lines on a coordinate plane. The goal is to find a common solution that satisfies all equations in the system simultaneously.
Such an occurrence happens at the point where the lines intersect.

• An equation might not be solvable alone if it has two variables, like in our original exercise with the equations for two lines.
• When two lines intersect, they have one unique solution as they cross each other on a graph.
• If the lines are parallel, they never meet, meaning there is no solution.

Understanding systems of equations helps us solve real-world problems where two conditions need to be satisfied at once. For instance, this could help plan budgets or calculate travel times based on different speeds. By finding the intersection point, we're identifying the exact conditions where both equations hold true together.

Solving Linear Equations

Solving linear equations is all about finding the value of the variables that make the equation true. Linear equations are algebraic expressions that form a straight line when graphed on a coordinate plane. Let's break down the process:

• First, we need to express the equation in terms of the variables involved. In our exercise, these are given as two equations relating to a single variable, \(y=3x+\frac{1}{2}\) and \(y=2x+1\).
• The next step involves isolating one variable so as to make it easier for solving the equation. This was done in our solution by setting the equations equal to each other and solving for \(x\).
• After finding \(x\), substitute this value back into one of the original equations to solve for \(y\).

Applying these steps to our exercise provided the result as the point \(\left( \frac{1}{2}, 2 \right)\). Solutions to linear equations can have no solutions, one solution, or infinitely many solutions depending on how many lines intersect. Understanding this helps discern possible outcomes, primarily when the equations pertain to practical scenarios.

Graphing Lines

Graphing lines on a coordinate grid can make it easier to visualize equations and see relationships between variables clearly. When graphing a linear equation, you can follow these steps:

• Recognize the slope-intercept form: \(y = mx + b\). Here, \(m\) is the slope, indicating the steepness and direction of the line, and \(b\) is the y-intercept, where the line crosses the y-axis.
• To graph a line, start by plotting the y-intercept on the graph.
• Use the slope to determine the next points on the line. The slope \(m\) is the ratio \(\Delta y/\Delta x\); for example, a slope of 3 means the line rises by 3 for every step to the right.

Both original equations in the exercise, \(y = 3x + \frac{1}{2}\) and \(y = 2x + 1\), can be graphed using these principles. By plotting the lines, you can visually determine where they intersect, confirming the solution of the point \(\left( \frac{1}{2}, 2 \right)\). Graphing not only helps verify algebraic solutions but also gives a clear picture of the equation's real-world applications.