Problem 26 - 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. $$\left\\{\begin{array}{rr}x+y= & 7 \\\2 x-3 y= & -1\end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \((4, 3)\).

Step 1: Label Equations

Label the given equations for reference. Let the first equation be \( (1) \) and the second equation be \( (2) \):\( (1) : x + y = 7 \)\( (2) : 2x - 3y = -1 \)

Step 2: Express One Variable in Terms of the Other

Solve equation \( (1) \) for \( x \):\[ x = 7 - y \]

Step 3: Substitute into the Second Equation

Substitute \( x = 7 - y \) into equation \( (2) \):\[ 2(7-y) - 3y = -1 \]

Step 4: Simplify Substituted Equation

Simplify the resulting equation:\[ 14 - 2y - 3y = -1 \]

Step 5: Combine Like Terms

Combine like terms on the left side:\[ 14 - 5y = -1 \]

Step 6: Solve for \( y \)

Rearrange the equation to solve for \( y \):\[ 14 + 1 = 5y \]\[ 15 = 5y \]\[ y = 3 \]

Step 7: Solve for \( x \)

Substitute \( y = 3 \) back into the expression for \( x \) from Step 2:\[ x = 7 - 3 \]\[ x = 4 \]

Step 8: Write the Solution in Ordered Pair Form

The solution to the system is the ordered pair: \( (x, y) = (4, 3) \). This pair satisfies both equations in the system.

Key Concepts

Linear Equations

Linear equations form the foundation of algebra and are crucial in understanding systems of equations. A linear equation is any equation that can be written in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. Some key characteristics of linear equations include:

• They graph as a straight line.
• The highest degree of the variable(s) is one.
• They can have one solution, no solution, or infinitely many solutions depending on how they're graphically represented.

In our exercise, we are given two such equations:

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

Both of these are linear equations because they meet all the criteria above. Understanding these properties helps in visualizing potential solutions, either through graphing or algebraic manipulation.

Substitution Method

The substitution method is a powerful technique used to solve systems of linear equations. It involves solving one of the equations for one variable and then substituting that expression into the other equation. Here’s a breakdown of the method applied in our example:

• **Solve one equation for one variable:** We first rearrange the first equation \( x + y = 7 \) to express \( x \) in terms of \( y \): \( x = 7 - y \).
• **Substitute this expression into the other equation:** We replace \( x \) with \( 7 - y \) in the second equation \( 2x - 3y = -1 \), resulting in \( 2(7 - y) - 3y = -1 \).
• **Solve for the remaining variable:** After simplification, we find that \( y = 3 \).
• **Substitute back to find the other variable:** Using \( y = 3 \) in \( x = 7 - y \), we get \( x = 4 \).

This method is efficient because it reduces the number of variables in one equation, simplifies problem-solving, and enhances understanding of how the solutions interact.

Ordered Pairs

In the context of a solution to a system of equations, ordered pairs represent the values of \( x \) and \( y \) that satisfy all equations in the system. An ordered pair is written as \( (x, y) \), meaning it lists the \( x \)-coordinate first and the \( y \)-coordinate second. Consider why ordered pairs are useful:

• They clearly define a single point in the coordinate plane.
• They help verify solutions by checking if substituting the values into each equation results in a true statement.
• They allow us to describe solutions visually (on graphs) or algebraically.

In this exercise, the solution \( (x, y) = (4, 3) \) meets all criteria as it satisfies both equations in the system:

• Substitute \( x = 4 \) and \( y = 3 \) into \( x + y = 7 \), resulting in \( 4 + 3 = 7 \), which is true.
• Substitute \( x = 4 \) and \( y = 3 \) into \( 2x - 3y = -1 \), resulting in \( 8 - 9 = -1 \), also true.

This ordered pair not only solves the system but demonstrates the connection between algebraic and geometric interpretations of solutions.