Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine any point(s) of intersection algebraically. Then verify your result numerically by creating a table of values for each function. $$\begin{aligned} &x-y=-4\\\ &x+2 y=5 \end{aligned}$$

Step-by-Step Solution

Answer

The point of intersection of the functions is \((1,3)\). The tables for \(x−y=−4\) and \(x+2y=5\), at \(x=1\) proved to be equal to \(y=3\), confirming the results.

Step 1: Solve one of the equations for one variable

Solve the first equation for \(x\). Rearrange the first equation to read: \(x = y - 4\).

Step 2: Substitute the solved equation into the second equation

Replace \(x\) from second equation with \(y - 4\) getting: \(y - 4 + 2y = 5\). From here, simplify the equation to find the value of \(y\).

Step 3: Solve for the other variable

Solve for \(y\) to get \(y = 3\). Now substiute \(y = 3\) into the first original equation to find the corresponding \(x\) value.

Step 4: Verify with a table

Once point of intersection is found, create tables for each equation and confirm if at \(x = 1\) each function yields a \(y\) value of 3.

Key Concepts

Point of Intersection

When two lines cross on a graph, the spot where they meet is the point of intersection.
For this problem, the point of intersection is where both equations share the same values of and for their variables.
To find it, we first need to determine the values of x and y that satisfy both equations:

• Equation 1: \( x - y = -4 \)
• Equation 2: \( x + 2y = 5 \)

Finding this point helps us understand where the relationships between the two variables overlap.
This point is essential in various scenarios since it indicates where two situations or datasets behave the same way.

Substitution Method

The substitution method is a common strategy used to solve systems of equations.
Here's a quick guide:

• Solve one equation for one variable.
• Substitute this expression into the other equation.
• Simplify and solve for the remaining variable.

In this exercise, we solved the first equation \( x = y + 4 \).
Next, replace "x" in the second equation with \( y + 4 \), leading to \( (y + 4) + 2y = 5 \).
Simplify this to find \( y = 3 \).
This method effectively reduces the number of variables, making it manageable to find their actual values.

Algebraic Verification

After finding the potential solution for the system, it's crucial to verify it to avoid errors.
Algebraic verification involves plugging the found values back into the original equations:

• Equation 1: Substitute \( x = 1 \) and \( y = 3 \): \( 1 - 3 = -4 \). It holds true.
• Equation 2: Substitute \( x = 1 \) and \( y = 3 \): \( 1 + 2(3) = 1 + 6 = 5 \). It holds true.

If both equations are satisfied, the solution \( (x, y) =(1, 3) \) is indeed correct.
This step ensures that no algebraic mistakes were made and confirms the result's accuracy.

Table of Values

Creating a table of values is a practical way to verify the solution by testing few x values.
You make columns for x and y, then plot each equation using simple substitutions.
In these tables:

• Equation 1: Choose x-values, calculate corresponding y by solving \( y = x + 4 \).
• Equation 2: Use the same x-values, find y by solving \( y = \frac{5-x}{2} \).

When you plug in x=1:

• Equation 1 gives y = 1 + 4 = 5
• Equation 2 gives y = (5-1)/2 = 2

We should get y=3 for x=1 to verify the solution, emphasizing aligning computations with the intersection point.
Tables visually confirm our solution, enhancing our confidence in its correctness.