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

Step 1: Identify the System of Equations

The given system consists of two equations: 1. \(x - y = 3\)2. \(x + 3y = 7\). Our objective is to solve for \(x\) and \(y\).

Step 2: Express x in terms of y

From the first equation, express \(x\) in terms of \(y\):\(x = y + 3\). This gives us a relationship between \(x\) and \(y\) that can be substituted into the second equation.

Step 3: Substitute x Into the Second Equation

Substitute \(x = y + 3\) from the first equation into the second equation \(x + 3y = 7\):\((y + 3) + 3y = 7\).

Step 4: Simplify and Solve for y

Simplify the equation:\(y + 3 + 3y = 7\) becomes \(4y + 3 = 7\).Subtract 3 from both sides to get \(4y = 4\).Divide both sides by 4 to find \(y = 1\).

Step 5: Find x Using y

Now, use \(y = 1\) in the expression \(x = y + 3\):\(x = 1 + 3 = 4\).

Step 6: Verify the Solution

Substitute \(x = 4\) and \(y = 1\) back into the original equations to verify:1. For \(x - y = 3\): \(4 - 1 = 3\), which is true.2. For \(x + 3y = 7\): \(4 + 3 \times 1 = 7\), which is also true.Both check out, confirming the solution is correct.

Key Concepts

Linear Equations

Linear equations are the foundation of algebra and involve variables raised to the first power. They create straight lines when graphed, hence the name "linear." A system of linear equations consists of two or more equations that share common variables. For example, the system \( x - y = 3 \) and \( x + 3y = 7 \) is a canonical instance where each equation describes a line in a coordinate plane. The goal when solving such a system is to find the values of the variables that satisfy all the equations simultaneously.

• Coefficients: Numbers multiplying the variables, like '1' in front of 'x' in our example.
• Constant terms: The standalone numbers, such as '3' and '7'.
• Solutions: Values like \( x = 4 \) and \( y = 1 \) which make each equation true.

Linear equations can intersect, be parallel, or even be the same line, driving different solution sets: a single point (one solution), all points along the line (infinitely many solutions), or no point at all (no solution).
Understanding these characteristics is key to effectively solving systems of linear equations.

Solution Verification

Solution verification is a critical step in solving equations. Once you find potential solutions, you must ensure they satisfy the original equations. This process checks for accuracy and consistency to confirm the solution's validity.
With our example system, we found \( x = 4 \) and \( y = 1 \). To verify:

• Substitute \( x = 4 \) and \( y = 1 \) into both equations.
• For the first equation \( x - y = 3 \): substitute to get \( 4 - 1 = 3 \), which holds true.
• For the second equation \( x + 3y = 7 \): substitute to get \( 4 + 3 \times 1 = 7 \), which also holds true.

When each substituted equation confirms the solution fits, you can confidently claim you've solved the system correctly.
Verifying ensures logical consistency and helps prevent errors often seen in algebraic manipulations.

Substitution Method

The substitution method is a powerful technique to solve systems of equations, especially when one equation is easily solvable for a variable. It involves solving one equation for one variable and substituting that expression into the other equation(s).
With our system:

• From \( x - y = 3 \), solve for \( x \): \( x = y + 3 \).
• Substitute \( x = y + 3 \) into \( x + 3y = 7 \): becomes \( (y + 3) + 3y = 7 \).
• Simplify and solve: \( 4y + 3 = 7 \), resulting in \( y = 1 \).
• Substitute back to find \( x \): Using \( x = y + 3 \), substitute \( y = 1 \), resulting in \( x = 4 \).

The substitution method simplifies systems by reducing the number of equations you deal with in each step and provides a straightforward path to finding the solution.