Problem 60 - Calculus Single Variable

Question

Find the point at which the lines determined by the two given equations intersect. $$ 3 x+5 y=14, x-y=2 $$

Step-by-Step Solution

Answer

The lines intersect at the point (3, 1).

Step 1: Express One Variable in Terms of the Other

Starting with the second equation, express \( x \) in terms of \( y \): \( x - y = 2 \). Add \( y \) to both sides to find \( x = y + 2 \). This allows us to substitute for \( x \) in the first equation.

Step 2: Substitute the Expression into the First Equation

Substitute \( x = y + 2 \) into the first equation: \( 3x + 5y = 14 \) becomes \( 3(y + 2) + 5y = 14 \). Simplify to obtain a single equation in terms of \( y \).

Step 3: Simplify the Equation

Expand \( 3(y + 2) + 5y = 14 \) to get: \( 3y + 6 + 5y = 14 \). Combine like terms to simplify to \( 8y + 6 = 14 \).

Step 4: Solve for \( y \)

Subtract 6 from both sides: \( 8y = 8 \). Divide both sides by 8 to solve for \( y \): \( y = 1 \).

Step 5: Find \( x \) Using \( y \)

Substitute \( y = 1 \) back into the expression for \( x \): \( x = y + 2 \). Therefore, \( x = 1 + 2 = 3 \).

Step 6: Write the Intersection Point

The point of intersection is the ordered pair \((x, y)\). Using the values found, the intersection point is \((3, 1)\).

Key Concepts

System of Equations

When dealing with a system of equations, you're working with multiple equations that share a set of unknowns. In real-world terms, this means having two or more lines on a graph and trying to find out where they meet or cross each other. For the problem at hand, our system consists of two equations:

• Equation 1: \(3x + 5y = 14\)
• Equation 2: \(x - y = 2\)

These equations describe two straight lines. We want to find the intersection of these lines, which represents the set of values \((x, y)\) that satisfy both equations simultaneously. Finding this point of intersection is crucial for understanding the relationship between these two lines visually and mathematically. Understanding systems of equations is not only about finding intersection points but also about comprehending the relationships between different algebraic representations.

Linear Equations

Linear equations are the simplest kind of algebraic equations, represented graphically as straight lines. They typically appear in the form \(ax + by = c\). Each line on a graph can be described by its slope and intercept. For our system:

• The first equation \(3x + 5y = 14\) can be rewritten to highlight its slope: \(y = -\frac{3}{5}x + \frac{14}{5}\).
• The second equation \(x - y = 2\) is straightforward to convert: \(y = x - 2\).

Linear equations are fundamental because they represent the steady, unchanging relationships and can predict how one variable influences another. When these lines are plotted on a graph, the point or points where they intersect are solutions to the system. Every intersection tells a story about collaboration or agreement represented by shared values for \(x\) and \(y\).

Substitution Method

The substitution method is a powerful tool for solving systems of equations. It involves solving one of the equations for one variable and substituting that expression into the other equation. This method is highly effective, especially when one equation is already simplified, or it can be easily manipulated to express one variable.
In our exercise:

• We started with the simpler equation \(x - y = 2\) and solved for \(x\), giving us \(x = y + 2\).
• This expression for \(x\) was then substituted into the first equation, transforming \(3x + 5y = 14\) into a solvable equation with just one variable.

By focusing on one variable at a time, we reduce the complexity of a problem into a single-variable equation, making it much simpler to handle. This method is not only efficient but also deepens your understanding of how equations relate to one another.

Solving for Variables

The ultimate goal in working with equations is to solve for unknown variables. After substitution, our equation turned into a straightforward algebraic equation. From here, solving for a variable means performing arithmetic operations to isolate that variable on one side of the equation.
In our example:

• The simplified equation, \(8y + 6 = 14\), required subtracting six from both sides to reach: \(8y = 8\).
• Then, dividing both sides by 8 revealed \(y = 1\)..
• Once \(y\) was found, substituting back allowed us to solve for \(x\), arriving at \(x = 3\).

Solving for variables is not just about finding specific numbers; it's about understanding the path to getting there and comprehending each step's purpose. The clarity gained through solving all parts enlightens how these numbers interact within equations, ultimately finding the point \((3, 1)\) where our lines intersect.