Problem 73 - Precalculus with Limits

Question

Think About It Consider the system of equations $$\left\\{\begin{array}{l}{a x+b y=c} \\ {d x+\epsilon y=f}\end{array}\right.$$ (a) Find values for \(a, b, c, d, e,\) and \(f\) so that the system has one distinct solution. (There is more than one correct answer.) (b) Explain how to solve the system in part (a) by the method of substitution and graphically. (c) Write a brief paragraph describing any advantages of the method of substitution over the graphical method of solving a system of equations.

Step-by-Step Solution

Answer

a) Possible values are \(a = 1, b = 1, c = 2, d = 2, e = -1, f = 1\). b) Solution by substitution gives \(x = 1, y = 1\). These are congruent when visually validated with a graph. c) Substitution method is more systematic and less impacted by scale of graph but can get complex, whereas graphical method is more visual but potentially less precise.

Step 1: Define a System of Equations

First, a system of equations that has one distinct solution needs to be defined. One possibility is to consider the equations \(x + y = 2\) and \(2x - y = 1\), so here \(a = 1, b = 1, c = 2, d = 2, e = -1\) and \(f = 1\).

Step 2: Solve by Substitution

To solve this system by substitution, isolate \(x\) in the second equation to get \(x = (1+y)/2\). Then, substitute this into first equation, which gives \((1+y)/2 + y = 2\). Solve this for \(y\), which gives \(y = 1\). Substituting \(y = 1\) into \(x = (1+y)/2\) gives \(x = 1\), so the solution is \(x = 1, y = 1\).

Step 3: Solve Graphically

To visualize the solution graphically, plot both equations. The intersection point of these two graphs represents the solution to the system of equations. Based on our calculation, this point should be at (1, 1).

Step 4: Analyse Methods

The substitution method can be more straightforward as it provides a systematic method for solution. A potential disadvantage is that it might be algebraically complex for more difficult problems. The graphical method, while intuitive, might be less precise, especially if exact values are not clear from the graph.

Key Concepts

Method of Substitution

The method of substitution is a powerful technique to solve systems of equations. It involves isolating one variable in one of the equations and then substituting the expression into the other equation. This allows you to solve for one variable and then back-solve for the other.

Here's a simple step-by-step:

• Choose one equation and solve for one variable in terms of the other.
• Substitute this expression into the other equation. This allows you to have just one equation with one variable.
• Solve this single-variable equation to find the value of the first variable.
• Once you have one variable, substitute it back into the expression you found earlier to find the second variable.

This method is particularly effective when one of the equations is easily solvable for one of the variables. It gives an exact solution, which makes it reliable when precision is needed. However, it can become cumbersome if the algebra gets too involved or if equations are not easily manipulable.

Graphical Method

The graphical method involves plotting both equations on a coordinate plane and visually identifying the intersection point(s). This point represents the solution to the system of equations since it satisfies both equations simultaneously.

Steps to solve using the graphical method:

• Convert each equation into the slope-intercept form (if necessary), which is \( y = mx + b \) where \( m \) is the slope and \( b \) is the y-intercept.
• Draw each line on a graph by starting at the y-intercept and using the slope.
• Find the point where the two lines intersect.
• Read off the coordinates of this intersection point to get the values of \( x \) and \( y \).

While this method is intuitive, especially for visual learners, it can be less precise when the intersection point has non-integer coordinates or if the graphs are not accurately drawn. It's excellent for understanding the relationship between the equations, but care must be taken to ensure accuracy.

Distinct Solution

A system of equations has a distinct solution when there is exactly one solution that satisfies both equations. This happens when the lines representing the equations intersect at one point on the graph.

For a system with two linear equations in two variables, conditions that ensure a distinct solution include:

• The coefficients should not be proportional, meaning the lines are not parallel.
• Both equations must represent two different lines that intersect.

A distinct solution is crucial in many applications because it provides a single set of values for the variables, which often represent key quantities in practical problems. In practice, this means having two reliable and independent pieces of information to solve a problem uniquely. This makes distinct solutions essential in fields like engineering, physics, economics, and more.