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

Step-by-Step Solution

Answer

The solution is \((-2, -12)\).

Step 1: Write Down the System of Equations

We are given the system of equations:\[ \begin{align*} 4x - 3y &= 28 \ 9x - y &= -6 \end{align*} \] Our goal is to find the values of \(x\) and \(y\) that satisfy both equations at the same time.

Step 2: Solve for One Variable

Start by solving the second equation for \(y\) in terms of \(x\): \[ y = 9x + 6 \] We will substitute this expression for \(y\) into the first equation.

Step 3: Substitute and Solve for x

Substitute \(y = 9x + 6\) into the first equation: \[ 4x - 3(9x + 6) = 28 \] Simplify and solve for \(x\): \[ \begin{align*} 4x - 27x - 18 &= 28 \ -23x - 18 &= 28 \ -23x &= 46 \ x &= -2 \end{align*} \] So the value of \(x\) is \(-2\).

Step 4: Substitute x Back to Find y

Now that we have \(x = -2\), substitute this value into the equation \(y = 9x + 6\): \[ y = 9(-2) + 6 \] Simplify to find \(y\): \[ y = -18 + 6 = -12 \] So the value of \(y\) is \(-12\).

Step 5: Write the Solution as an Ordered Pair

The solution to the system of equations, if it exists, is expressed as an ordered pair. Thus, the solution is \((-2, -12)\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that graph as straight lines. They typically take the form \( ax + by = c \), where \(a\), \(b\), and \(c\) are constants, and \(x\) and \(y\) are variables. These equations are fundamental in algebra because they are straightforward to manipulate.
Linear equations have several characteristics:

• They feature variables raised only to the first power.
• The solutions can be graphed as straight lines on the coordinate plane.
• Intercepts can often be directly determined, providing easy graphing points.

Understanding linear equations is crucial for more complex mathematical concepts like systems of equations, as they form the building blocks of these systems. With practice, solving linear equations becomes a quick and intuitive process.

Substitution Method

The substitution method is a strategy for solving systems of linear equations. It involves solving one of the equations for one variable and substituting that expression into the other equation. This technique simplifies the problem by reducing the number of variables:

• Solve for one variable in terms of the other using one of the equations.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation.
• Substitute back to find the value of the other variable.

This method is particularly useful when one of the equations is easily manipulated to isolate one variable. It helps reduce complex equations into simpler forms, making it more manageable to find the solution.

Solving Systems

Solving systems of equations entails finding values for the variables that satisfy all equations in the system simultaneously. For example, given the system:\[ \begin{align*}4x - 3y &= 28 \9x - y &= -6\end{align*}\] we are tasked with finding where these two linear equations intersect on a graph, signifying the solution.There are multiple methods to solve systems of equations, including:

• Graphing: Plotting both equations to find their intersection point.
• Substitution Method: Employing substitution to solve sequentially like in our provided example.
• Elimination Method: Adding or subtracting equations to eliminate a variable and solve for the remaining one.

Success in solving systems is about finding which method fits best with the given equations, leading to a solution like \((-2, -12)\).

Ordered Pairs

Ordered pairs are a way to express solutions to systems of equations, typically written in the form \((x, y)\). This notation shows the specific values for \(x\) and \(y\) that make each equation in the system true
In the context of graphing:

• The \(x\)-value indicates the position on the horizontal axis.
• The \(y\)-value indicates the position on the vertical axis.

For example, the solution \((-2, -12)\) means if you plot a point at \(x = -2\) and \(y = -12\), it lies on both lines represented by the given equations. Ordered pairs provide a concise way to represent solutions, especially when graphing or communicating them to others.