Problem 33 - Precalculus Mathematics for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered pair form given in Example 6. $$\left\\{\begin{array}{r}3 x+2 y=8 \\\x-2 y=0\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is (2, 1).

Step 1: Analyze the System of Equations

We have a system of linear equations: 1. \(3x + 2y = 8\)2. \(x - 2y = 0\).To solve the system, we can use either the substitution or elimination method. Let's choose substitution for this problem.

Step 2: Solve the Second Equation for x

From equation 2, \(x - 2y = 0\), solve for \(x\):\[x = 2y\].This expression for \(x\) will be used in the first equation to find the value of \(y\).

Step 3: Substitute the Expression for x into the First Equation

Substitute \(x = 2y\) into the first equation, \(3x + 2y = 8\):\[3(2y) + 2y = 8\].Simplify the equation:\[6y + 2y = 8\].

Step 4: Solve for y

Combine like terms in the equation:\[8y = 8\].Divide both sides by 8:\[y = 1\].Now we have the value of \(y\).

Step 5: Find the Value of x using y = 1

Substitute \(y = 1\) back into \(x = 2y\) to find \(x\):\[x = 2(1)\]\[x = 2\].

Step 6: Write the Solution as an Ordered Pair

The solution to the system is the ordered pair \((x, y)\). We found \(x = 2\) and \(y = 1\), thus the solution is:\((2, 1)\).

Key Concepts

Substitution Method

The substitution method is an effective technique for solving systems of linear equations. This method involves expressing one variable in terms of another and then substituting this expression into the other equation.
It simplifies the system into a single equation with one variable.

• Step-by-Step Process: Begin by solving one of the equations for one variable in terms of the other. This is usually easiest when one of the equations is already close to an isolated form, like the second equation in our exercise: \(x - 2y = 0 \).
• Here, solving for \(x\) gives us \(x = 2y \). This equation expresses \(x\) entirely in terms of \(y\).
• Next, substitute this expression into the other equation, which allows us to work with just one equation and one variable.

Using substitution makes solving equations more straightforward, especially when one equation is easier to manipulate. Make sure to carefully track and simplify as you progress through the steps.

Elimination Method

The elimination method, also known as the addition method, is another technique used to solve systems of linear equations. Unlike substitution, elimination focuses on adding or subtracting equations to eliminate a variable outright.

• Purpose: The goal of elimination is to remove one variable by aligning the equations such that adding or subtracting removes the variable.
• This is beneficial when both equations can easily be manipulated with coefficients that make one variable's coefficients opposites or equal.
• For instance, if you had equations like \(3x + 2y = 8\) and \(x - 2y = 0\), you could add them directly because the \(y\) terms are opposites, effectively eliminating \(y\). This leaves behind a new equation containing only \(x\).
• Then, solve the resulting equation for the remaining variable, and substitute it back into one of the original equations to find the other variable.

Although not used in our specific problem, understanding the elimination method provides a flexible approach when substitution isn't intuitive.

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics used to express the solution of a system of equations. When you solve a system, each solution can be expressed as \((x, y)\), where \(x\) and \(y\) are specific values that satisfy each equation in the system.

• Format: The ordered pair denotes the coordinates of a point in a coordinate plane. The first element is the \(x\) coordinate, while the second is the \(y\) coordinate.
• In the given exercise, after solving the system, we found the ordered pair \((2, 1)\). This indicates that at \(x = 2\) and \(y = 1\), both equations \(3x + 2y = 8\) and \(x - 2y = 0\) hold true.
• Ordered pairs are not only used in systems of linear equations but also in determining relationships and mappings between sets.

These pairs make it easier to plot and visualize solutions on a graph, providing a tangible representation of where equations intersect.