Problem 12 - Precalculus: Mathematical 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 3. $$\left\\{\begin{aligned} x+y &=7 \\ 2 x-3 y &=-1 \end{aligned}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Determine which method to use

Examine the given system of equations: \[ \begin{aligned} x+y &= 7 \ 2x-3y &= -1 \end{aligned} \] to determine which method to use to solve it. In this case, the Substitution Method seems appropriate because we can easily solve for one variable in one of the equations.

Step 2: Solve Equation 1 for x

Start by solving the first equation for \( x \). From \( x+y=7 \), we can express \( x \) in terms of \( y \) as: \[ x = 7 - y \]

Step 3: Substitute expression for x in Equation 2

Substitute \( x = 7 - y \) into the second equation \( 2x-3y = -1 \). This gives: \[ \begin{aligned} 2(7-y) - 3y &= -1 \ 14 - 2y - 3y &= -1 \end{aligned} \]

Step 4: Simplify and solve for y

Combine like terms in the equation: \[ 14 - 5y = -1 \]. Solve for \( y \) by isolating the variable: \[ \begin{aligned} 14 + 1 &= 5y \ 15 &= 5y \ y &= 3 \end{aligned} \]

Step 5: Substitute back to find x

Use the value of \( y \) to find \( x \). Substitute \( y = 3 \) back into \( x = 7 - y \): \[ x = 7 - 3 = 4 \]

Step 6: Write the solution in ordered-pair form

The solution to the system is the ordered pair \((x, y)\). In this case, that pair is \((4, 3)\).

Step 7: Verify the solution

Substitute \((4, 3)\) back into the original equations to check: \[ \begin{aligned} 4 + 3 &= 7 \quad \text{(True)} \ 2(4) - 3(3) &= -1 \quad \text{(True)} \end{aligned} \]. Since both equations are satisfied, the solution is verified.

Key Concepts

Substitution Method

The substitution method is an algebraic technique used to solve systems of equations. It involves expressing one variable in terms of another using one of the equations, then substituting this expression into the other equation. This method is exceptionally useful when one of the equations can be easily rearranged to solve for a single variable.
To effectively use the substitution method:

• Choose an equation that can be easily solved for one variable. In our example, the equation \( x + y = 7 \) is perfectly set up to express \( x \) in terms of \( y \).
• Solve for that variable, creating an expression that relates the two variables. From \( x + y = 7 \), we get \( x = 7 - y \).
• Substitute this expression into the other equation. By substituting \( x = 7 - y \) into the second equation \( 2x - 3y = -1 \), the system is reduced to a single equation with one variable.

Through these steps, you can find solutions systematically and verify correctness.

Ordered-Pair Solution

An ordered-pair solution is a way of expressing the solution to a system of equations. It represents the values of variables that satisfy both equations in the system. In our example, the ordered-pair solution is recovered as \((4, 3)\).
This means:

• \( x = 4 \)
• \( y = 3 \)

The solution \((4, 3)\) needs to hold true for every original equation when substituted back. This means
substituting \((4, 3)\) into the initial system should not alter the truth of the equalities. It's essentially the intersection point of two lines represented by the equations on a graph, denoting where both conditions meet.
When working with ordered-pair solutions:

• Verify by substituting back into original equations, which ensures the solution is correct.
• Remember that if the system has infinitely many solutions, they would be represented in a similar pair format, describing a line of intersecting solutions.

Solving Linear Equations

Solving linear equations involves finding the values of variables that make the equation true. In the context of a system of equations, we seek a solution or solutions that satisfy all equations simultaneously.
In our original system:

• The equation \( x + y = 7 \) implies a line where any \( (x, y) \) combinations satisfy this condition.
• The equation \( 2x - 3y = -1 \) signifies another line, and the intersection of these lines represents the solution.

When solving such equations:

• Use algebraic methods like substitution, elimination, or graphing for solution finding.
• Often, simplifying one equation helps address the system more effectively.

Our solution consists of systematically finding where these lines meet. In the world of algebra, visually think of it as solving where paths cross.
Every solution should be verified by ensuring it fulfills each equation in its original form. This approach is central to verifying and understanding systems of linear equations.