Problem 32 - College Algebra and Calculus: An Applied Approach

Question

Solve the system by elimination Then state whether the system is consistent inconsistent. $$\left\\{\begin{array}{l}0.02 x-0.05 y=-0.19 \\ 0.03 x+0.04 y=0.52\end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = 8\) and \(y = 7\). After substitution of the values of \(x\) and \(y\) into both equations, it is observed that both equations are satisfied. Therefore, the system of equations is consistent.

Step 1: Multiply both equations to get rid of decimals and have a common coefficient

Multiply the first equation by 3 and the second equation by 2. This will give: \[\begin{align*} 0.06x - 0.15y = -0.57 \0.06x + 0.08y = 1.04 \end{align*}\]Finally, simplify the above to get:\[\begin{align*} 6x - 15y = -57 \6x + 8y = 104 \end{align*}\]

Step 2: Subtract the second equation from the first one to eliminate x

Subtracting the equations, we get:\[\begin{align*} (6x - 15y) - (6x + 8y) = -57 - 104 \ -23y = -161\end{align*}\]Doing the subtraction, \(x\) gets eliminated, leaving an equation with only \(y\).

Step 3: Solve for y

Divide the equation by -23 to isolate \(y\):\[\begin{align*} -23y = -161 \y = 7\end{align*}\]

Step 4: Substitute y = 7 into one of the original equations to solve for x

Let's pick the first equation:\[\begin{align*} 6x - 15*(7) = -57 \6x = 48 \x = 8\end{align*}\]

Step 5: Checking the Solution

Substitute \(x = 8\) and \(y = 7\) into the two original equations. If both equations are satisfied, then the solution is consistent. If not, the system is inconsistent.

Key Concepts

Elimination Method

The elimination method is a popular technique for solving systems of linear equations. It involves strategically eliminating one variable to solve for the other. This method works by combining the equations in such a way that one variable cancels out, leaving a simpler equation with only one unknown.

To use this method effectively, follow these steps:

• Multiply or divide one or both equations, if necessary, to obtain the same coefficient for one of the variables across the equations.
• Add or subtract the equations to eliminate one of the variables.
• Solve the resulting equation for the remaining variable.
• Substitute the solution back into one of the original equations to find the other variable.

This method is not only straightforward but also efficient, especially when the coefficients of one of the variables are already aligned. However, it requires careful manipulation of the equations to avoid errors.

Consistent and Inconsistent Systems

Understanding whether a system of equations is consistent or inconsistent is crucial to solving it. A consistent system has at least one solution, which means the lines representing the equations intersect at least at one point on a graph.

Here's how to determine the nature of the system:

• Consistent and Independent: If the system has exactly one solution, it is consistent and independent. In this case, the lines intersect at a single point.
• Consistent and Dependent: If the system has infinitely many solutions, it is consistent and dependent. This occurs when the lines representing the equations coincide.
• Inconsistent: If the system has no solution, it is inconsistent. This happens when the lines are parallel and do not intersect.

To verify whether the given system is consistent or inconsistent, check if the values of the variables satisfy all the original equations. If they do, the system is consistent. Otherwise, it's inconsistent.

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra. These equations take the form of a straight line when graphed. The solution to a linear equation is the set of values that make the equation true.

To solve linear equations, follow these general steps:

• Identify the equation that needs solving and determine which variable appears easiest to isolate.
• Use arithmetic operations to isolate the variable on one side of the equation.
• Once the variable is isolated, solve the arithmetic to find its value.
• Plug this value back into the original equation to verify the correctness of the solution.

Linear equations may come in single or multiple forms, such as in a system where they must be considered collectively. Mastery of solving these basic units lays the groundwork for tackling more complex algebraic problems involving multiple equations and variables.