Problem 18 - Precalculus: Mathematical 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 3. $$\left\\{\begin{array}{rr} 0.2 x-0.2 y= & -1.8 \\ -0.3 x+0.5 y= & 3.3 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution is \((-6, 3)\).

Step 1: Write the system of equations

The given system of equations is: \[\begin{align*} 0.2x - 0.2y &= -1.8 \ -0.3x + 0.5y &= 3.3 \end{align*}\] The goal is to solve this system either by finding a single solution, showing that there is no solution, or that there are infinitely many solutions.

Step 2: Eliminate Decimals

To simplify the calculations, multiply both equations by 10 to eliminate the decimals. \[\begin{align*} 2x - 2y &= -18 \ -3x + 5y &= 33 \end{align*}\] Now the equations have integer coefficients which are easier to handle.

Step 3: Multiply to Eliminate a Variable

Let's eliminate the variable \(x\). Multiply the first equation by 3 and the second equation by 2: \[\begin{align*} 6x - 6y &= -54 \ -6x + 10y &= 66 \end{align*}\]

Step 4: Add Equations to Solve for \(y\)

Add the two new equations from Step 3 to eliminate \(x\): \[\begin{align*} (6x - 6y) + (-6x + 10y) &= -54 + 66 \end{align*}\] This simplifies to: \[\begin{align*} 4y &= 12 \end{align*}\] Divide by 4 to solve for \(y\): \[\begin{align*} y &= 3 \end{align*}\]

Step 5: Substitute \(y = 3\) to Find \(x\)

Substitute \(y = 3\) back into one of the original equations, for instance, \(2x - 2y = -18\): \[\begin{align*} 2x - 2(3) &= -18 \end{align*}\] Simplify and solve for \(x\): \[\begin{align*} 2x - 6 &= -18 \ 2x &= -12 \ x &= -6 \end{align*}\]

Step 6: Write the Solution as an Ordered Pair

The solution to the system is the point where the two lines intersect. We found \(x = -6\) and \(y = 3\), so the solution in ordered-pair form is: \((-6, 3)\). This is the intersection point of the two lines, confirming that the system has one unique solution.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same sets of variables. In the provided exercise, we are looking at a system involving two equations with the variables \(x\) and \(y\):

• \(0.2x - 0.2y = -1.8\)
• \(-0.3x + 0.5y = 3.3\)

These equations are linear, meaning they graph as straight lines when plotted on a coordinate plane. The objective when solving a system of linear equations is to determine the specific values of \(x\) and \(y\) that satisfy both equations at the same time. This is equivalent to finding the point at which the two lines intersect. If they intersect at one point, the system has a unique solution. If they are parallel, there are no solutions, and if they overlap completely, there are infinitely many solutions.

Ordered Pair

The solution to a system of linear equations, when it exists, is often expressed as an ordered pair. An ordered pair is a set of values for the variables, represented as \((x, y)\).In the context of our exercise, solving the system led us to the solution \((-6, 3)\). This means that substituting \(x = -6\) and \(y = 3\) into both equations of the original system satisfies both equations, demonstrating that the point of intersection where these two linear equations meet is at \((-6, 3)\).Expressing the solution as an ordered pair provides a clear and concise way to communicate the intersection point of the two lines represented by the original system.

Solving Linear Systems

Solving linear systems means finding the values of variables that satisfy all equations simultaneously. There are several methods to solve linear systems, like substitution, elimination, or using matrices. The exercise utilizes the elimination method to find the solution.Here's a quick recap of the process:

• First, make the equations easier to work with by eliminating decimals. This was done by multiplying each part by 10.
• Next, we eliminated one variable by manipulating the equations to have opposite coefficients for \(x\).
• After eliminating \(x\), we solved the resulting equation for \(y\).
• Once \(y\) was found, its value was substituted back into one of the original equations to find \(x\).

This systematic approach efficiently narrows down the possible values for the variables, ensuring that the solution \((-6, 3)\) uniquely satisfies both equations. Always ensure to check your solution by substituting the values back into the original equations to verify their correctness, confirming that this ordered pair is indeed the intersection of the two lines.