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

Step-by-Step Solution

Answer

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

Step 1: Write the system of equations

We are given the system of equations: \( x + y = 7 \) and \( 2x - 3y = -1 \). Our goal is to find values for \( x \) and \( y \) that satisfy both equations.

Step 2: Solve the first equation for y

From the first equation \( x + y = 7 \), we can express \( y \) in terms of \( x \): \( y = 7 - x \).

Step 3: Substitute y into the second equation

Substitute \( y = 7 - x \) into the second equation \( 2x - 3y = -1 \). This gives us \( 2x - 3(7 - x) = -1 \).

Step 4: Simplify the equation

Simplify the substituted equation: \( 2x - 21 + 3x = -1 \), which simplifies further to \( 5x - 21 = -1 \).

Step 5: Solve for x

Add 21 to both sides to get \( 5x = 20 \). Then, divide both sides by 5 to solve for \( x \): \( x = 4 \).

Step 6: Substitute x back into the expression for y

Use the expression for \( y \) from Step 2: \( y = 7 - x \). Substitute \( x = 4 \): \( y = 7 - 4 = 3 \).

Step 7: Write the solution as an ordered pair

The solution to the system of equations is \( (4, 3) \).

Step 8: Verify the solution

Substitute \( x = 4 \) and \( y = 3 \) back into both original equations to verify.\( x + y = 7 \Rightarrow 4 + 3 = 7 \text{ (True)} \) and \( 2x - 3y = -1 \Rightarrow 2(4) - 3(3) = 8 - 9 = -1 \text{ (True)} \). Thus, the solution is verified.

Key Concepts

Substitution Method

The substitution method is a technique for solving systems of linear equations. It involves solving one equation for one variable and then using that expression to substitute into the other equation. This helps eliminate one of the variables so that we can solve for the other.

For instance, let's consider the system of equations:

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

In the substitution method, we begin by solving one of these equations for one variable. Here, we can solve Equation 1 for \(y\):

• \(y = 7 - x\)

This expression for \(y\) is then substituted into Equation 2, replacing every instance of \(y\). The resulting equation will only have the variable \(x\), which can then be solved using traditional algebra techniques.

Once \(x\) has been found, we substitute it back into the expression we found for \(y\) to find the corresponding \(y\) value. This method ensures a systematic approach to isolating variables and effectively solves linear systems.

Ordered Pairs

An ordered pair is a two-element collection commonly used to represent coordinates on a plane or solutions to systems of linear equations. Ordered pairs are usually written as \((x, y)\). In the context of systems of equations, each ordered pair represents a possible solution where both coordinates satisfy every equation in the system.

In the provided exercise, after using the substitution method, we calculated the values \(x = 4\) and \(y = 3\). This results in the ordered pair \((4, 3)\), which is the point where the graphs of the two equations intersect.

These ordered pairs are crucial because they provide a concise representation of the solution. It's essential to remember that solutions to systems of equations can be single ordered pairs, no solution, or infinitely many solutions if the lines represented by the equations coincide.

Verification of Solutions

Verification of a solution in a system of linear equations ensures that the calculated values satisfy all original equations. This involves substituting the found values back into each equation and checking if both sides of the equation are equal.

For our system:

• Solution as an ordered pair: \((4, 3)\)
• Substitute into first equation: \(x + y = 7\)
• Verification: \(4 + 3 = 7\), which holds true.

For the second equation:

• Equation: \(2x - 3y = -1\)
• Substitute: \(2(4) - 3(3) = 8 - 9 = -1\)
• This calculation checks out as \(-1 = -1\).

Verifying solutions is a critical step, as it confirms the accuracy of the solution. This process is particularly helpful in ensuring that no algebraic mistakes occurred during calculations, guaranteeing that the solution is indeed correct.