Problem 30 - 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. $$\left\\{\begin{aligned} -4 x+12 y &=0 \\ 12 x+4 y &=160 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The solution is \((x, y) = (12, 4)\).

Step 1: Understand the system of equations

We have two equations forming a linear system: 1. \(-4x + 12y = 0\) 2. \(12x + 4y = 160\).Our goal is to find the values of \(x\) and \(y\) that satisfy both equations at the same time. We also need to determine if there is one solution, no solution, or infinitely many solutions.

Step 2: Simplify both equations

Divide the first equation by -4 and the second equation by 4 to simplify them.Equation 1: \(-4x + 12y = 0 \rightarrow x - 3y = 0\)Equation 2: \(12x + 4y = 160 \rightarrow 3x + y = 40\).

Step 3: Express one variable terms from one equation

From the simplified first equation \(x - 3y = 0\), solve for \(x\):\(x = 3y\).

Step 4: Substitute the expression into the second equation

Substitute \(x = 3y\) into the simplified second equation \(3x + y = 40\):\(3(3y) + y = 40\).

Step 5: Solve for \(y\)

Simplify the equation from the previous step:\(9y + y = 40\) \(10y = 40\) \(y = 4\).

Step 6: Substitute back to find \(x\)

Use \(y = 4\) in the equation \(x = 3y\):\(x = 3(4)\) \(x = 12\).

Step 7: Solution verification

To ensure the solution is correct, substitute \(x = 12\) and \(y = 4\) back into the original equations. Check Equation 1: \(-4(12) + 12(4) = -48 + 48 = 0\) is true. Check Equation 2: \(12(12) + 4(4) = 144 + 16 = 160\) is true.Both hold true, confirming the solution.

Key Concepts

Linear equation

Linear equations are equations of the first degree, meaning they have no exponents greater than 1. They are called 'linear' because they graph as straight lines on a coordinate plane. In solving a system of linear equations, we look for values of the variables that satisfy all equations in the system simultaneously.

Each equation is typically in the form of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. The coefficients \(a\) and \(b\) dictate how steep or flat the line is, while the constant \(c\) shifts the line.

• The objective is to find the point(s) of intersection, which represent common solutions to all equations.
• If two lines intersect at only one point, there is a unique solution.
• Two parallel lines indicate no common solution unless they are the same line, which denotes infinitely many solutions.

Solution verification

Solution verification is crucial to ensure that the calculated solution indeed satisfies the original system of equations. After solving the equations, always substitute the values back into the original equations to confirm their correctness. This step ensures that no algebraic errors have been made in the process, which could lead to incorrect results.

In the context of the given system, we verified the solution by substituting \(x = 12\) and \(y = 4\) back into both original equations. Both substitutions confirmed their respective identities, proving that the solution is correct.

• Equation 1 Verification: Substituted values yield \(-4(12) + 12(4) = 0\), which simplifies to \(0 = 0\).
• Equation 2 Verification: Substituted values result in \(12(12) + 4(4) = 160\), which simplifies correctly to \(160 = 160\).

This process guarantees that the values satisfy each equation and hence fully solve the system.

Substitution method

The substitution method is a common technique used to solve systems of linear equations. It involves isolating one variable in one of the equations and substituting its expression into the other equation. This approach simplifies the system from two variables across two equations to a single equation with one variable.

In our exercise, we began by solving the simplified equation \(x - 3y = 0\) for \(x\), resulting in \(x = 3y\). We then substituted this expression into the second equation \(3x + y = 40\). This step involves replacing \(x\) with \(3y\):

• Simplified Equation: \(3(3y) + y = 40\)
• Further Simplification: \(9y + y = 40\) simplifies to \(10y = 40\)
• Solving for \(y\) provides \(y = 4\)

Then, substituting \(y = 4\) back to find \(x\) yields \(x = 3(4) = 12\). By replacing one variable with its equivalent, the substitution method simplifies solving a system of equations and systematically reaches a solution.