Problem 30 - Algebra and Trigonometry

Question

\(21-48=\) 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 \(6 .\) $$ \left\\{\begin{array}{l}{-4 x+12 y=0} \\ {12 x+4 y=160}\end{array}\right. $$

Step-by-Step Solution

Answer

The solution is \((12, 4)\).

Step 1: Simplify First Equation

The first equation is \(-4x + 12y = 0\), which can be simplified by dividing all terms by 4, resulting in \(-x + 3y = 0\).

Step 2: Solve for x in Terms of y

From the equation \(-x + 3y = 0\), solve for \(x\) by rearranging: \(x = 3y\).

Step 3: Substitute x in Second Equation

Substitute \(x = 3y\) into the second equation: \(12(3y) + 4y = 160\), which simplifies to \(36y + 4y = 160\).

Step 4: Simplify and Solve for y

Combine the \(y\) terms: \(40y = 160\). Solve for \(y\) by dividing both sides by 40, resulting in \(y = 4\).

Step 5: Substitute y Back to Find x

Substitute \(y = 4\) back into \(x = 3y\), yielding \(x = 3(4) = 12\).

Step 6: Confirm the Solution

Check the solution \((x, y) = (12, 4)\) in the original equations to verify correctness. Both \(-4(12) + 12(4) = 0\) and \(12(12) + 4(4) = 160\) hold true, confirming the solution.

Key Concepts

Linear Equations

Linear equations are equations that graph as straight lines. They have variables raised to the power of one and are represented in the form of \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our example exercise, we have two linear equations:

• \(-4x + 12y = 0\)
• \(12x + 4y = 160\)

Linear equations can be solved using methods such as substitution, elimination, or graphing. Identifying whether a solution exists, or if there are infinitely many, is crucial in understanding systems of equations. Linear equations are foundational in algebra, helping us find intersections between lines, among many other applications. To solve a system of equations, the equations must first be analyzed and simplified if necessary.

Substitution Method

The substitution method is a technique to solve systems of equations by solving one equation for a single variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations is easy to solve for one of the variables.
For the exercise at hand, the substitution method involved solving the first simplified equation \(-x + 3y = 0\) for \(x\), resulting in \(x = 3y\). This expression for \(x\) was then substituted into the second equation, \(12x + 4y = 160\), helping us to eliminate \(x\) and solve for \(y\).
Steps involved in the substitution method include:

• Simplifying one of the equations, if necessary, to make substitution easier.
• Solving that equation for one variable.
• Substituting the expression from the solved equation into the other equation.
• Solving the resultant equation for the remaining variable.
• Substituting back to find the values of both variables.

It's a powerful method, allowing for systematic approaches to finding the solution.

Solution Verification

After finding a potential solution to a system of equations, verifying it is a crucial step. Solution verification ensures that the proposed values of the variables satisfy the original equations. In the exercise, the found solution was checked by plugging \(x = 12\) and \(y = 4\) back into the original equations.

• For \(-4x + 12y = 0\), substituting gives \(-4(12) + 12(4) = -48 + 48 = 0\).
• For \(12x + 4y = 160\), substituting results in \(12(12) + 4(4) = 144 + 16 = 160\).

Both results confirm the correctness of the solution. Verifying a solution is especially important to catch potential errors or oversights in calculations. It ensures accuracy and reliability in solving systems of equations.

Ordered Pairs

In the context of solving systems of equations, an ordered pair \((x, y)\) is used to express the solution. The ordered pair represents the intersection point of the two linear equations on a graph. It gives us specific values for \(x\) and \(y\), showing where both equations are true simultaneously.
For this exercise, the solution was identified as the ordered pair \((12, 4)\). This means when \(x = 12\) and \(y = 4\), both equations are satisfied. Ordered pairs are not only crucial in solutions but also in graphing and understanding relationships between variables. In some systems of equations, ordered pairs can represent multiple solutions, particularly when dealing with infinite solutions.
In our example, however, the ordered pair \((12, 4)\) is the unique solution, demonstrating the precise point of intersection of the two lines given by the equations.