Problem 124 - Elementary Algebra

Question

Solve the system of equations \(\left\\{\begin{array}{l}x+y=10 \\\ x-y=6\end{array}\right.\) (a) by graphing. (b) by substitution. (c) Which method do you prefer? Why?

Step-by-Step Solution

Answer

Graph: Intersection at (8, 2). Substitution: \(x = 8, y = 2 \). Prefer substitution for precision.

Step 1: Graphing - Step 1: Plot the first equation

Rewrite the first equation in slope-intercept form: \(x + y = 10 \rightarrow y = 10 - x\). Plot this line on a graph by finding the intercepts. The y-intercept is 10 (point (0,10)) and the x-intercept is 10 (point (10,0)).

Step 2: Graphing - Step 2: Plot the second equation

Rewrite the second equation in slope-intercept form: \(x - y = 6 \rightarrow y = x - 6\). Plot this line on the graph by finding the intercepts. The y-intercept is -6 (point (0,-6)) and the x-intercept is 6 (point (6,0)).

Step 3: Graphing - Step 3: Find the intersection

Observe where the two lines intersect on the graph. The point of intersection represents the solution to the system of equations.

Step 4: Substitution - Step 1: Solve one equation for one variable

From the first equation \(x + y = 10\), solve for \(y\): \(y = 10 - x\).

Step 5: Substitution - Step 2: Substitute into the second equation

Substitute \(y = 10 - x\) into the second equation \(x - y = 6\): \(x - (10 - x) = 6\).

Step 6: Substitution - Step 3: Solve for x

Simplify the equation: \(x - 10 + x = 6 \rightarrow 2x - 10 = 6 \rightarrow 2x = 16 \rightarrow x = 8\).

Step 7: Substitution - Step 4: Solve for y

Substitute \(x = 8\) back into \(y = 10 - x\): \(y = 10 - 8 = 2\). The solution is \(x = 8, y = 2\).

Step 8: Preference

The preferred method can vary; substitution often provides a more straightforward algebraic solution without needing to graph, making it easier to see the exact solution.

Key Concepts

Understanding Graphing of Linear Equations

Graphing is a visual way to solve a system of linear equations.
Start by converting the equations into slope-intercept form, which is y = mx + b. This makes it easier to plot the lines on a graph.
For instance, if we consider the equations from the problem, \(x + y = 10\) transforms to \(y = 10 - x\), and \(x - y = 6\) transforms to \(y = x - 6\).
Next, identify the intercepts:
For \(y = 10 - x\), the y-intercept is 10 (where x=0), and the x-intercept is 10 (where y=0).
For \(y = x - 6\), the y-intercept is -6 (where x=0), and the x-intercept is 6 (where y=0).
Graph these lines on the coordinate plane. The point where they cross is the intersection, representing the solution to the system.
Graphing provides a visual understanding and immediately shows where the solution lies.

Using the Substitution Method for Solving Systems

The substitution method is an algebraic approach.
Start by solving one of the equations for one variable.
In our problem, we take \(x + y = 10\) and solve for \(y\), which gives us \(y = 10 - x\).
Then, substitute \(y = 10 - x\) into the other equation \(x - y = 6\).
This substitution changes the second equation into one variable: \(x - (10 - x) = 6\).
Simplify and solve for \(x\):
\ x - 10 + x = 6 \ \Rightarrow 2x - 10 = 6 \ \Rightarrow 2x = 16 \ \Rightarrow x = 8\.
Now, substitute \(x = 8\) back into the first modified equation \(y = 10 - x\) to get \(y = 2\).
The solution is the coordinate pair (8, 2).
Substitution is often preferred for precise solutions, especially when graphing is not practical.

Finding the Intersection

In both methods, the intersection represents the solution to the system of equations.
Graphically, it is the point where two lines meet.
In our problem, plotting both equations shows that the lines intersect at (8, 2).
This intersection is confirmed algebraically through substitution.
The point (8, 2) is the common solution, meaning both equations are satisfied when \(x = 8\) and \(y = 2\).
Understanding the concept of intersection helps visualize solutions and confirms algebraic methods.