Problem 68 - 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. $$\begin{aligned} &4 x-y=4\\\ &x-4 y=1 \end{aligned}$$

Step-by-Step Solution

Answer

The point of intersection of the given equations is (1,0). This result was confirmed numerically.

Step 1: Algebraic Solution - Substitution method (Step 1)

First, rearrange one of the equations by solving for x or y and call this new equation (1). Let's consider the first equation \[4x - y = 4\]. If we re-arrange this equation to solve for y, we have \[y = 4x - 4\], which is our \(equation 1\).

Step 2: Algebraic Solution - Substitution method (Step 2)

Next, substitute \(equation 1\) into the second equation (Substitute y as 4x - 4 in x - 4y = 1), this gives us \[x - 4(4x - 4) = 1\], After solving this equation simplifying the expression on left, we get \[-15x = -15\]. Thus, solving for x we obtain x = 1

Step 3: Algebraic Solution - Finding y-value (Step 3)

Using the value of x = 1 from step 2, we substitute it into equation 1 to find the corresponding y-coordinate. Thus y = 4*(1) - 4 = 0. Therefore, the point of intersection is (1, 0).

Step 4: Numerical Verification - Creating Tables of Values (Step 4)

Take several values for x and calculate the corresponding y values using both equations. For equation 1: \(4x - y = 4\)| x | y ||----|----|| 1 | 0 || 2 | 4 || 3 | 8 | For equation 2: \(x - 4y = 1\)| x | y ||----|----|| 1 | 0 || 2 | 0.25 || 3 | 0.5 |Both tables indicate that the x-value of 1 corresponds to a y-value of 0 in both functions at the same time, confirming the algebraic solution.

Key Concepts

Substitution Method

The substitution method is an algebraic approach to solve a system of equations. It's particularly useful when you have two equations and want to find the values of two variables, like x and y. Here's how it works:

• Start by solving one of the equations for one variable. In this example, from the equation \(4x - y = 4\), we solve for \(y\) to get \(y = 4x - 4\). Let's call this new equation "equation 1."
• Next, substitute this expression for \(y\) in the other equation. So, in the second equation \(x - 4y = 1\), substitute \(y\) with \(4x - 4\). This simplifies the equation to \(x - 4(4x - 4) = 1\).
• Solving this simplified equation allows us to find the value of \(x\). In this example, it simplifies to \(-15x = -15\), giving us \(x = 1\).
• Finally, use this value of \(x\) in your earlier expression, \(y = 4x - 4\), to find \(y\). Therefore, \(y = 4 \cdot 1 - 4 = 0\).

The solution \((1, 0)\) is the intersection of the two equations using substitution.

Intersection Points

Intersection points in a system of equations are places where the graphs of the equations meet on the coordinate plane. They represent solutions where both equations are true simultaneously.

• Each equation in a system can be visualized as a line (if linear) on a graph.
• The point where these lines cross is an intersection point.
• For the given system \(4x - y = 4\) and \(x - 4y = 1\), the intersection point is found to be \((1, 0)\) through algebraic manipulation.

Understanding intersection points helps in visualizing solutions and validating the outcomes algebraically. In graphical terms, these are the coordinates where two curves or lines coincide.

Verification Method

When you find a solution to a system of equations, it's crucial to verify its correctness. One reliable way is creating a table of values for each equation to check consistency.

• Create a table of \(x\) and \(y\) pairs for each equation using several \(x\) values.
• In these tables, see if there's a common pair of \(x, y\) values that satisfies both equations simultaneously.
• In our solved example, both equations have the common pair \((1, 0)\), verifying the solution obtained earlier.

By comparing these tables, you numerically confirm that the algebraic solution is correct. If the values match, you’ve verified your solution.

Algebraic Solutions

Algebraic solutions involve solving equations using techniques from algebra, such as substitution, elimination, or graphing for verification.

• The algebraic approach seeks to eliminate variables, either by substitution or rearranging equations, to find explicit values.
• In the example \(4x - y = 4\) and \(x - 4y = 1\), you derive the intersection point by first solving for a variable and then substituting back into the original equation.
• This method is efficient and reduces the risk of graphical misinterpretations that occur when plotting by hand.

Besides finding solutions, this method provides a clear pathway of calculations, enhancing understanding of the relationships between variables in equations.