Problem 1 - Calculus for Biology and Medicine

Question

Solve each linear system of equations. In addition, for each system, graph the two lines corresponding to the two equations in a single coordinate system and use your graph to explain your solution. $$ \begin{array}{l} x-y=1 \\ x-2 y=-2 \end{array} $$

Step-by-Step Solution

Answer

The solution is \(x = 4\) and \(y = 3\). The lines intersect at (4, 3).

Step 1: Understand the Given Equations

We have two linear equations: \(x - y = 1\) and \(x - 2y = -2\). Our task is to find the values of \(x\) and \(y\) that satisfy both equations simultaneously.

Step 2: Express \(x\) from the First Equation

Start by isolating \(x\) in the first equation. From \(x - y = 1\), we get: \(x = y + 1\).

Step 3: Substitute into the Second Equation

Substitute \(x = y + 1\) into the second equation \(x - 2y = -2\): \( (y + 1) - 2y = -2\). This simplifies to \(-y + 1 = -2\).

Step 4: Solve for \(y\)

Rearrange the equation \(-y + 1 = -2\) to find \(y\): Subtract 1 from both sides: \(-y = -3\). Multiply through by -1: \(y = 3\).

Step 5: Solve for \(x\)

Use \(y = 3\) in the expression derived in Step 2: \(x = y + 1\). Thus, \(x = 3 + 1 = 4\).

Step 6: Solution Verification

Verify the solution by substituting \(x = 4\) and \(y = 3\) back into the original equations. **First equation:** \(4 - 3 = 1\). **Second equation:** \(4 - 2(3) = -2\). Both statements are true, confirming the solution.

Step 7: Graph the Equations

Graph each equation on the coordinate plane to visually verify the solution. - For \(x - y = 1\): Rearrange to \(y = x - 1\). - For \(x - 2y = -2\): Rearrange to \(y = \frac{1}{2}x + 1\). Both lines intersect at \((4, 3)\), confirming the solution.

Key Concepts

Solving Systems of Equations

When you have two or more equations working together, that's a system of equations. The challenge is to find a common pair of values for variables that satisfy each equation. It's like finding a treasure that fits perfectly into multiple treasure chests! In the context of our exercise, we have two equations:

• \(x - y = 1\)
• \(x - 2y = -2\)

We must find the values of \(x\) and \(y\) that make both equations true at the same time. There are various methods to tackle this kind of problem:

• The Substitution Method
• The Elimination Method
• Graphical Method

All methods can lead to the solution.
In our solution, we use substitution:

1. Isolate \(x\) in the first equation, yielding \(x = y + 1\) .
2. Substitute this into the second equation, simplifying to find \(y = 3\).
3. With \(y\) known, substitute back to find \(x\), resulting in \(x = 4\).

These values \(x = 4\) and \(y = 3\) make both original equations true, confirming the solution!

Graphing Linear Equations

Graphing is a fantastic visual approach to understanding linear equations. It helps us see relationships and verify solutions. A linear equation is expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This explains how steep the line is and where it crosses the y-axis.

For our equations:

• The first equation \(x - y = 1\) transforms to \(y = x - 1\).
• The second equation \(x - 2y = -2\) transforms to \(y = \frac{1}{2}x + 1\).

Each equation gives us a unique line on the coordinate plane. The slope tells us the direction and angle of each line:

• \(y = x - 1\) has a slope of 1, rising steeply.
• \(y = \frac{1}{2}x + 1\) rises more gently with a slope of \(\frac{1}{2}\).

Plotting these on a graph allows us to see exactly how they differ and where they might intersect.

Intersection of Lines

The intersection point of two lines is key. This is where both lines meet or cross on the graph, indicating the solution to the system of equations. It's the spot where the solutions for both equations match perfectly.

To find this let’s observe our two plotted lines:

• \(y = x - 1\)
• \(y = \frac{1}{2}x + 1\)

When graphed, these lines intersect at the point \((4, 3)\). What does this mean?
It means that at the point \((4, 3)\), both conditions given by the equations are satisfied.

• The first equation becomes true since \(4 - 3 = 1\).
• The second equation also holds as \(4 - 2 \times 3 = -2\).

The intersection confirms our algebraic solution, making this a very reliable check. Graphical methods are powerful for interpretation and can convey solutions in a way that equations alone cannot.