Problem 26 - Precalculus

Question

Solve each system of linear equations by elimination. $$\begin{array}{r} 3 x+2 y=6 \\ -3 x+6 y=18 \end{array}$$

Step-by-Step Solution

Answer

The solution is \( x = 0 \) and \( y = 3 \).

Step 1: Add the Equations

The system of equations is: \( 3x + 2y = 6 \) and \( -3x + 6y = 18 \). To eliminate \(x\), add the first equation to the second equation: \[(3x + 2y) + (-3x + 6y) = 6 + 18 \] The resulting equation is \( 8y = 24 \).

Step 2: Solve for y

With the equation from Step 1, \( 8y = 24 \), solve for \( y \) by dividing both sides by 8: \[y = \frac{24}{8} = 3\] So, \( y = 3 \).

Step 3: Substitute y into One Equation

Substitute \( y = 3 \) back into the first original equation to solve for \( x \): \[3x + 2(3) = 6\] Simplify to get: \[3x + 6 = 6\] Subtract 6 from both sides: \[3x = 0\]

Step 4: Solve for x

From the equation \( 3x = 0 \), solve for \( x \) by dividing both sides by 3: \[x = \frac{0}{3} = 0\] So, \( x = 0 \).

Step 5: Conclusion

The solution to the system is \( x = 0 \) and \( y = 3 \). This means the two lines intersect at the point \( (0, 3) \).

Key Concepts

Elimination Method

The elimination method is a popular way to solve systems of linear equations. This technique involves adding or subtracting equations to eliminate one of the variables, making it easier to solve the system.
In our original exercise, we start with two equations. The goal is to remove one of the variables, either \( x \) or \( y \), from the equations. By manipulating the equations, particularly by aligning them in a way that the coefficients of one of the variables are opposites, we can add or subtract the equations to eliminate that variable.

• This method is useful when the equations are already set up to cancel out one variable easily.
• Eliminating a variable simplifies the system into a single equation, making it straightforward to find the value of the remaining variable.
• Once a variable is eliminated, the resulting equation can be solved using basic algebraic operations.

In short, the elimination method is favored for its straightforward approach to simplifying systems of linear equations into a much easier-to-solve format.

Solving for Variables

Once we have simplified the system of equations using the elimination method, we proceed to solve for the unknown variables. Solving for variables involves isolating the variable on one side of the equation to determine its value.
For example, in our step-by-step solution, after eliminating \( x \), we find ourselves with the equation \( 8y = 24 \). Solving for \( y \) involves performing arithmetic operations to isolate \( y \). Dividing both sides by 8 gives \( y = 3 \).

• First, simplify the equation as much as possible.
• Perform operations that will leave the variable on one side, usually achieved by addition, subtraction, multiplication, or division.
• Remember, whatever operation you apply to one side of the equation, apply it to the other to maintain equality.

After determining the value of one variable, substitute it back into one of the original equations to find the second variable, as we did by substituting \( y = 3 \) to find \( x = 0 \).
This ensures that both solutions are consistent with the original system of equations.

Intersection of Lines

Once both variables \( x \) and \( y \) are determined, interpreting the solution involves understanding where the two lines intersect on the coordinate plane. In the context of linear equations, the solution represents the point of intersection.
In our problem, finding \( x = 0 \) and \( y = 3 \) means the two lines intersect at the point \( (0, 3) \). This point of intersection is significant as it signifies that both equations are satisfied simultaneously at this coordinate pair.

• The intersection point is where the lines cross and represents the solution to the system of equations.
• If the lines are not parallel and not the same, there is exactly one intersection point.
• Understanding the geometric interpretation of these solutions helps in visualizing the problem and verifying the accuracy of the solution.

In summary, the intersection point obtained from solving systems of linear equations illustrates the common solution shared by both equations, providing insight into their geometric relationship.