Problem 23 - 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. $$\left\\{\begin{array}{l} 2 x-3 y=9 \\ 4 x+3 y=9 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has a single solution: (3, -1).

Step 1: Set up the system of equations

We have the system of equations: \[\begin{array}{l} 2x - 3y = 9 \ 4x + 3y = 9 \end{array}\] We will solve these equations to determine if there is one solution, no solution, or infinitely many solutions.

Step 2: Add the equations to eliminate y

We add the two equations to eliminate the variable \(y\). This gives us: \[(2x - 3y) + (4x + 3y) = 9 + 9\] \[6x = 18\] This simplifies to \(x = 3\).

Step 3: Substitute x back to find y

Substitute \(x = 3\) into the first equation \(2x - 3y = 9\): \[2(3) - 3y = 9\] \[6 - 3y = 9\] Subtract \(6\) from both sides: \[-3y = 3\] Divide by \(-3\): \(y = -1\).

Step 4: Verify the solution

Substitute \(x = 3\) and \(y = -1\) into the second equation \(4x + 3y = 9\): \[4(3) + 3(-1) = 9\] \[12 - 3 = 9\] \[9 = 9\] Since both equations are satisfied, \((3, -1)\) is a solution to the system.

Key Concepts

Linear Equations

Linear equations are mathematical statements that express a relationship between two variables, typically denoted as \(x\) and \(y\), involving constants and the first power of the variables. In their simplest form, they appear as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Linear equations graph as straight lines on the Cartesian plane.

In a system of linear equations, we seek values for these variables that satisfy all equations simultaneously. Solving such systems often involves methods like substitution or elimination.

• If the lines intersect at a point, there is one unique solution.
• If they are parallel, there is no solution.
• If they coincide, there are infinitely many solutions.

Understanding the geometric interpretation of these solutions can be very insightful. For instance, in our exercise, the system involves two linear equations. Solving it gives us exactly one point, indicating the lines intersect at exactly one location.

Solution Verification

Verifying the solution to a system of equations is crucial. It confirms that the calculated values indeed satisfy all the given equations. This process involves substituting the solution back into the original equations to ensure consistency.

For our exercise, we found \(x = 3\) and \(y = -1\). Verification means substituting these values into each equation:

• First equation: \(2x - 3y = 9\)
• Second equation: \(4x + 3y = 9\)

Substituting \(x = 3\) and \(y = -1\) into both equations showed that each equation balanced, thus confirming the solution is correct.

This practice of solution verification not only assures accuracy but also helps in developing a deeper understanding of the solution's validity. It is a critical step you should never skip in solving systems of equations.

Substitution Method

The substitution method is a popular technique for solving systems of linear equations. It involves solving one of the equations for a single variable and then substituting that expression into the other equation(s).

In our exercise, once the equations were simplified by elimination, we found \(x = 3\). The next step involved substituting \(x = 3\) back into one of the original equations, specifically \(2x - 3y = 9\), to find \(y\).

Here is how it looked:

• Substitute \(x = 3\) into \(2x - 3y = 9\), giving \(6 - 3y = 9\).
• Solving for \(y\) led to \(y = -1\).

This method systematically helps in breaking down the problem into more manageable parts, making it easier to find the values of variables one at a time. The substitution method is especially effective when one equation is simple and directly solvable for one of the variables.