Problem 22 - 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. $$\left\\{\begin{array}{l}x-y=3 \\\x+3 y=7\end{array}\right.$$

Step-by-Step Solution

Answer

The system has one solution: \((4, 1)\).

Step 1: Understand the System

We are given a system of two linear equations: 1. \(x - y = 3\) 2. \(x + 3y = 7\)

Step 2: Solve the First Equation for x

Rearrange the first equation \(x - y = 3\) to solve for \(x\). This gives:\[ x = y + 3 \]

Step 3: Substitute x into the Second Equation

Substitute the expression from Step 2 into the second equation \(x + 3y = 7\). We replace \(x\) with \(y + 3\):\[ (y + 3) + 3y = 7 \]

Step 4: Simplify and Solve for y

Simplify the equation from Step 3:\[ y + 3 + 3y = 7 \] Combine like terms:\[ 4y + 3 = 7 \] Subtract 3 from both sides:\[ 4y = 4 \] Divide by 4:\[ y = 1 \]

Step 5: Back-Substitute to Find x

Now that \(y\) is found, substitute \(y = 1\) back into the expression for \(x\) from Step 2:\[ x = 1 + 3 \] \[ x = 4 \]

Step 6: Write the Solution as an Ordered Pair

The solution to the system of equations is the pair \((x, y) = (4, 1)\).

Key Concepts

Linear Equations

Linear equations are the foundation of many mathematical concepts you'll encounter. In simple terms, a linear equation is an algebraic equation that forms a straight line when graphed. These equations usually have constant coefficients and variables that are raised only to the first power. For example, in the equation \(x - y = 3\), both \(x\) and \(y\) are variables we need to solve for. The defining feature of linear equations is that they don't have exponents other than 1, and they don't multiply the variables by each other.In the context of a system of equations, we are dealing with two or more linear equations at the same time. Our goal is often to find a common solution, which means a set of values for the variables that satisfy all equations involved. It's like finding the balance where both equations agree.

Substitution Method

The substitution method is a straightforward technique for solving systems of equations. It involves replacing one variable with an expression obtained from another equation. This is effective because it reduces the number of equations and unknowns we need to handle at one time.Here's how it works in practice:

• First, select an equation and express one variable in terms of the other. For example, from \(x - y = 3\), you can express \(x\) as \(x = y + 3\).
• Next, substitute this expression into the other equation. This means replacing the variable you've just expressed with the new expression. In our example, substitute \(x = y + 3\) into \(x + 3y = 7\), which leads to a single equation \((y + 3) + 3y = 7\).
• The substitution method simplifies a system of linear equations to a single equation in one variable, making it easier to solve. Once this is solved, back-substitute to find the value of the other variable.

This method is intuitive and often mirrors how we intuitively handle unknowns in everyday problem-solving scenarios.

Ordered Pair Solutions

An ordered pair solution is a way to express the answer to a system of equations. Once you've found the values for the variables, these are typically written in the form \((x, y)\). This format clearly displays which value corresponds to each variable.Consider the solution \((4, 1)\) from our exercise. This ordered pair means that \(x = 4\) and \(y = 1\) fit both original equations: \(x - y = 3\) and \(x + 3y = 7\).Ordered pairs also have a deeper meaning. They represent a point on a coordinate plane where two lines intersect. This point is where both equations hold true simultaneously. Understanding this helps to visualize how algebra can translate into graphing concepts, showing the intersection of solutions in a concrete way. Ordered pairs are not just numbers; they show the meeting point of linear relationships.