Problem 5 - Thinking Mathematically

Question

In Exercises 5-12, solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}x+y=6 \\ x-y=2\end{array}\right.\)

Step-by-Step Solution

Answer

The solution to the system of equations is the intersection point (4, 2).

Step 1: Rearrange Equations to y=mx+b Form

To plot the lines onto a graph, it is easiest to use the slope-intercept form of a linear equation, which is \(y = mx + b\). For the given equations, this will translate into: \(\left\{\begin{array}{l}y=6-x \ y=x-2\end{array}\right.\)

Step 2: Graph the Equations

On the same axis, graph both equations. The line \(y = 6 - x\) is a downward sloping line that intersects the y-axis at 6. The line \(y = x - 2\) is an upward sloping line that intersects the y-axis at -2. The intersection of these two lines will give the solution to the system of equations.

Step 3: Find Intersection Point

Identify the exact coordinates of the intersection point of the two lines by solving the two equations simultaneously. This is achieved by setting \(6 - x = x - 2\) to yield \(x = 4\). Substituting \(x = 4\) into either of the original equations gives \(y = 2\). Therefore, the intersection point is (4, 2).

Step 4: Validate the Solution

To validate the solution, these coordinates need to be plugged back into both original equations. For the first equation, \(x + y = 6\), substituting in \(x = 4\) and \(y = 2\) results in \(6 = 6\). Similarly, for the second equation, \(x - y = 2\), substituting gives \(2 = 2\). Both equations are satisfied, validating the solution.

Key Concepts

Graphing Systems

When solving a system of equations through graphing, we are trying to find the point where two lines intersect on a coordinate plane. This method involves translating the algebraic problem into a visual picture, which can be helpful for understanding relationships between the equations.
Graphing systems involve the following steps:

• Convert each equation into a graphable form, such as the slope-intercept form, which is easier to plot.
• Draw each line on the same coordinate plane.
• Observe where the two lines meet, which represents the solution.

Graphing provides a visual way to discover the solution that might not be as apparent through algebraic manipulation alone. It's a great way to see the relationship between the equations and their solutions visually.

Linear Equations

Linear equations are the foundation of graphing systems. These equations can be written in the form \(y = mx + b\), where ***m*** is the slope of the line, and ***b*** is the y-intercept. The slope represents the steepness of the line, while the y-intercept is where the line crosses the vertical axis.
Linear equations form straight lines when plotted on a graph. The main characteristics are:

• Slope ***m***: It shows how much ***y*** changes for a given change in ***x***.
• Intercept ***b***: Marks the starting point of the line on the y-axis.

This simple form allows for easily graphing each line, by determining just a couple of points or using the slope and the intercept directly.

Intersection Point

The intersection point in graphing systems of equations is critical as it provides the solution. For the equations \(y = 6 - x\) and \(y = x - 2\), the intersection represents the values of ***x*** and ***y*** that satisfy both equations simultaneously.
To find the intersection point:

• Set the equations equal to each other if both are in the form \(y = mx + b\), resulting in a single equation such as \(6 - x = x - 2\).
• Solve for ***x***, giving you one coordinate of the intersection point.
• Replace ***x*** in one of the original equations to find the corresponding ***y*** value.

The intersection point (4, 2) in this example signifies that when ***x = 4***, the value of ***y*** must be 2 for both equations to hold true.

Solution Validation

Solution validation is a crucial last step in solving a system of equations. Once the intersection point is identified, it is important to ensure that it satisfies both original equations to confirm that it is correct.
To validate the solution:

• Substitute the identified ***x*** and ***y*** values back into the original equations.
• Check if these values satisfy each equation, ensuring both sides are equal.

In our case, substituting \((4, 2)\) into \(x + y = 6\) yields \(6 = 6\), and into \(x - y = 2\) yields \(2 = 2\). Both equalities hold true, confirming the solution is correct. This step is essential to avoid mistakes and verify the accuracy of the result obtained from the graphing method.