Problem 8 - Technical Mathematics with Calculus

Question

Solve each system of equations by addition-subtraction, or by substitution. Check some by graphing. $$\begin{aligned} &x-y=7\\\ &3 x+y=5 \end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations is \( x = 3 \), \( y = -4 \), which can be checked graphically by observing that both lines intersect at the point (3, -4).

Step 1: Express y in terms of x from the first equation

Isolate variable y in the first equation to prepare for substitution. Add y to both sides and then subtract 7 from both sides to get: \( y = x - 7 \).

Step 2: Substitute the expression for y into the second equation

Replace y in the second equation with the expression found in Step 1 to get: \( 3x + (x - 7) = 5 \).

Step 3: Combine like terms and solve for x

Combine the x terms and add 7 to both sides to solve for x: \( 4x = 12 \), then divide by 4 to get \( x = 3 \).

Step 4: Substitute the value of x into the expression for y

Now that we have \( x = 3 \), substitute it into \( y = x - 7 \) to find the value of y: \( y = 3 - 7 = -4 \).

Step 5: Check the solution by graphing

Graph both equations on the same coordinate plane and observe if they intersect at the point (3, -4).

Key Concepts

Addition-Subtraction Method

The addition-subtraction method, also known as the elimination method, is a technique used to solve a system of linear equations. This method involves adding or subtracting the equations to eliminate one of the variables, making it easier to solve for the remaining variable.

For example, in the given exercise, we can eliminate the variable y by adding the two equations:

• (1) x - y = 7
• (3) 3x + y = 5

When we add equation (1) and equation (3) together, the y terms cancel each other out, leaving us with:

• 4x = 12

From here, we can solve for x by dividing both sides by 4, which gives us x = 3. Once we have the value of x, we can substitute it back into either of the original equations to find the value of y. This method is particularly effective when the coefficients of one of the variables are opposites or can easily be made opposites.

Substitution Method

The substitution method is another strategy for solving systems of equations. This method involves solving one of the equations for one variable in terms of the other variable, and then substituting that expression into the other equation. This process will yield an equation in one variable, which can then be solved.

In the exercise given, we isolate y in the first equation (1):

• y=x−7

Next, we substitute this expression for y into the second equation (3):

• 3x+(x−7)=5

Now, with an equation in one variable, we solve for x. After finding the value of x, we substitute it back into the first equation to find y. This method can be particularly useful when one of the equations is already solved for one variable or can be easily manipulated to solve for one variable.

The key to the substitution method is to ensure that the substitution is correctly applied and that the variable isolated at first is the one that makes substitution straightforward and simple.

Graphical Method

The graphical method of solving systems of equations involves plotting each equation on a graph and finding the point where the two lines intersect. This point of intersection represents the solution to the system, as it satisfies both equations simultaneously.

For our exercise, we would graph the following two equations:

• x−y=7
• 3x+y=5

The graph of each equation is a straight line. Once the equations are graphed, we look for the intersection point, which in this case is (3, -4). This confirms our solutions obtained by the substitution method.

The graphical method is particularly useful as a visual check. By graphing the equations, we can see whether there is one solution (the lines intersect at a single point), no solution (the lines are parallel and never intersect), or infinitely many solutions (the lines coincide). Additionally, graphing provides a visual understanding of the relationship between the variables in the system.