Problem 14 - 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. $$\left\\{\begin{array}{l} 4 x-3 y=28 \\ 9 x-y=-6 \end{array}\right.$$

Step-by-Step Solution

Answer

The system has one solution: \((-2, -12)\).

Step 1: Eliminate One Variable

To begin, we'll use the elimination method to eliminate one variable. Multiply the second equation by 3 to align the coefficients of \(y\).Original equations:\(\begin{align*}4x - 3y & = 28 \9x - y & = -6\end{align*}\)Multiply the second equation by 3:\(\begin{align*}27x - 3y & = -18\end{align*}\)

Step 2: Subtract Equations

Now, subtract the modified second equation from the first equation to eliminate \(y\).\(\begin{align*}(4x - 3y) - (27x - 3y) & = 28 - (-18) \4x - 3y - 27x + 3y & = 46 \-23x & = 46\end{align*}\)

Step 3: Solve for x

Solve the equation from step 2 for \(x\).\(\begin{align*}-23x & = 46 \x & = \frac{46}{-23} \x & = -2\end{align*}\)

Step 4: Substitute x in One Equation

Substitute \(x = -2\) back into one of the original equations to find \(y\). Let's use the first equation.\(\begin{align*}4(-2) - 3y & = 28 \-8 - 3y & = 28 \-3y & = 28 + 8 \-3y & = 36 \y & = -12\end{align*}\)

Step 5: Write the Solution as an Ordered Pair

The solution to the system of equations is the ordered pair \( (x, y) = (-2, -12)\).

Key Concepts

Elimination Method

The elimination method is a powerful tool for solving systems of equations. This method involves strategically adding or subtracting equations to eliminate one of the variables. By doing so, you're simplifying the problem to a single-variable equation, making it easier to solve step by step.

In the given system:

• First, align the coefficients of one of the variables in both equations. In our example, we multiplied the second equation by 3 so that the coefficients of \(y\) become the same magnitude (\(-3\) and "\(+3\)").
• Next, subtract one equation from the other. This process will cancel out \(y\), and you'll be left with an equation containing only \(x\).
• Solve for \(x\). Once \(x\) is found, you can then use the substitution method to find the other variable.

The elimination method is very efficient, especially when the coefficients of a variable are easily alignable through simple multiplication.

Substitution Method

After using the elimination method to find one variable, the substitution method steps in to help find the other.

Once one of the variables has been determined (in this case, \(x = -2\)), we use that value in one of the original equations to solve for the second variable.

• Substitute the solved value back into one of the original equations. Both equations are valid for substitution, but you might choose the simpler one for ease of calculations.
• In our scenario, substituting \(x = -2\) into the first equation, \(4x - 3y = 28\), we simplify to find \(y\).
• This leads to the expression \(-3y = 36\), resulting in \(y = -12\).

The substitution method complements the elimination method well, ensuring a complete solution to the system of equations.

Ordered Pairs

In systems of equations, the solution is typically expressed as an ordered pair, \((x, y)\). An ordered pair is a simple way to represent the point of intersection of the two lines described by the equations on a graph.

For our system, the coordinates \((-2, -12)\) specify this intersection point where both equations are met simultaneously.

• The first number in the ordered pair, \(-2\), represents the \(x\)-coordinate.
• The second number, \(-12\), signifies the \(y\)-coordinate.

Ordered pairs are crucial in systems of equations because they provide an intuitive understanding of solutions, making it clear where two or more equations meet in a graphical context.