Problem 8 - Algebra and Trigonometry

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

Step-by-Step Solution

Answer

The solution is \((4, 1)\).

Step 1: Analyze the System of Equations

We have two equations forming a system: \(x-y=3\) and \(x+3y=7\). Our goal is to find values for \(x\) and \(y\) that satisfy both equations simultaneously. We will use either the substitution method or elimination method to solve this system.

Step 2: Solve for One Variable

We can solve the first equation for \(x\): \(x = y + 3\). This expression will allow us to substitute for \(x\) in the second equation.

Step 3: Substitute into the Second Equation

Substitute \(x = y + 3\) into the second equation: \((y + 3) + 3y = 7\). This simplifies to \(y + 3 + 3y = 7\).

Step 4: Simplify and Solve for y

Combine like terms in the equation: \(4y + 3 = 7\). Subtract 3 from both sides to isolate terms with \(y\): \(4y = 4\). Divide both sides by 4 to solve for \(y\): \(y = 1\).

Step 5: Substitute Back to Find x

Use the expression for \(x\) from Step 2 (\(x = y + 3\)). Substitute \(y = 1\) to find \(x\): \(x = 1 + 3 = 4\).

Step 6: Verify the Solution

Check whether \((4, 1)\) satisfies both original equations. Plugging into \(x - y = 3\), we get \(4 - 1 = 3\), which is correct. For \(x + 3y = 7\), we get \(4 + 3\times1 = 7\), which is also correct. Thus, \((4, 1)\) is a solution to the system.

Key Concepts

Understanding the Substitution Method

The substitution method is a straightforward way to solve a system of equations. It involves expressing one variable in terms of the other in one of the equations and substituting this expression into the other equation. This reduces the system to an equation with only one variable, making it easier to solve.

Let's look at our system:

• Equation 1: \(x-y=3\)
• Equation 2: \(x+3y=7\)

We begin by isolating \(x\) in Equation 1, leading to \(x = y + 3\).
This expression for \(x\) allows us to substitute into Equation 2. Therefore, wherever we see \(x\) in Equation 2, we replace it with \(y + 3\), yielding a single variable equation: \((y + 3) + 3y = 7\).

Once we substitute and simplify the equation, we can solve for \(y\). Then, we use the result and substitute back into the expression for \(x\) to find its value. This method is particularly useful when the system is simple, or when one equation easily isolates a variable.

Elimination Method Explained

The elimination method, unlike substitution, involves adding or subtracting equations to eliminate one of the variables altogether. By aligning terms and adjusting coefficients when necessary, this method simplifies the system to a single equation with one variable.

In our case, we could transform the equations \(x-y=3\) and \(x+3y=7\) in such a way that adding or subtracting them directly cancels out one of the variables.
If we multiply the entire first equation by 3, we get \(3x-3y=9\). Next, subtract the second equation \(x + 3y = 7\) from \(3x-3y=9\):

• Equation: \(3x - 3y - (x + 3y) = 9 - 7\)
• Simplifies to: \(2x = 2\)

By solving the new equation \(2x = 2\), we find \(x = 1\), and substitute back into one of the original equations to solve for \(y\).
This method can be faster for systems where variables are easily cancelled when equations are combined.

Importance of Solution Verification

Verifying the solution of a system of equations is a crucial step that ensures accuracy and correctness. Once we've obtained potential solutions for \(x\) and \(y\), we need to substitute these values back into the original equations to confirm they satisfy both equations simultaneously.

Let's verify for our solution \((x, y) = (4, 1)\):

• Substitute into Equation 1: \(4 - 1 = 3\), which is correct.
• Substitute into Equation 2: \(4 + 3(1) = 7\), which is also correct.

Both equations are satisfied, confirming \((4, 1)\) is indeed the correct solution. Verification serves as a critical double-check, especially in more complex systems.