Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by substitution. $$\begin{array}{c}2 x-7 y=8 \\\\-3 x+\frac{21}{2} y=5\end{array}$$

Step-by-Step Solution

Answer

The system has no solution; it's inconsistent.

Step 1: Solve for x in terms of y

We start with the first equation: \[2x - 7y = 8\]We need to express \(x\) in terms of \(y\). Adding \(7y\) to both sides, we have:\[2x = 7y + 8\]Now, divide every term by 2:\[x = \frac{7y + 8}{2}\]

Step 2: Substitute x in the Second Equation

Now, substitute \(x = \frac{7y + 8}{2}\) into the second equation:\[-3x + \frac{21}{2}y = 5\]Replacing \(x\), the equation becomes:\[-3\left( \frac{7y + 8}{2} \right) + \frac{21}{2}y = 5\]Multiply through to eliminate the fraction:\[-\frac{3(7y + 8)}{2} + \frac{21}{2}y = 5\]

Step 3: Simplify the Substituted Equation

Multiply \(-3\) with the numerator inside the parentheses:\[-\frac{21y + 24}{2} + \frac{21}{2}y = 5\]Simplifying within the fraction gives:\[-\frac{21y}{2} - \frac{24}{2} + \frac{21}{2}y = 5\]This simplifies further to:\[-\frac{21y}{2} - 12 + \frac{21}{2}y = 5\]

Step 4: Solve for y

Notice that \(-\frac{21y}{2}\) and \(\frac{21}{2}y\) cancel each other out, leaving:\[-12 = 5\]This is incorrect, indicating this system has no solution. Thus, this system of equations is inconsistent.

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of equations. It involves isolating one variable in one equation and then substituting that expression into the other equation. This strategy reduces the system to a single equation with one unknown variable, which is typically easier to solve.

• Step 1: Solve for one variable in one of the equations. For instance, if you have the equation \(2x - 7y = 8\), you could solve for \(x\) to express it in terms of \(y\).
• Step 2: Substitute this expression into the other equation. By replacing \(x\)'s value with the expression you derived, this step leads to an equation with only one variable (\(y\) in this case).
• Step 3: Simplify and solve the single-variable equation. Once \(y\) is found, you can easily determine \(x\) by back-substitution into your initial expression.

This method comes in handy especially when equations involve simple linear relationships or when you're solving by hand. Choosing the simpler equation to isolate a variable initially can save time and reduce errors. Always re-evaluate your solution by plugging back the values into the original equations for verification.

Inconsistent System

In an inconsistent system of equations, no set of values for the variables will satisfy all the equations simultaneously. Such systems naturally cannot have solutions. In the context of linear equations, inconsistency often arises due to the lines being parallel in a graph, meaning they do not intersect at any point.

• When simplifying your substituted equation results in a false statement like \(-12 = 5\), it signals that the system is inconsistent.
• This typically happens when the coefficients of the variables are proportional, yet the constant terms differ. For example, two equations may have the same slope but different intercepts.
• Graphically, these equations appear as distinct, non-intersecting lines, confirming that there’s no common solution.

Understanding when a system is inconsistent is crucial because it saves unnecessary time trying to find non-existent solutions. Always confirm your findings with both the algebraic method and, if possible, by plotting the equations on a graph.

Solve by Substitution

Solving systems of equations by substitution involves replacing one variable with an expression from another equation. This approach lets you systematically reduce and simplify systems, which is useful in finding exact solutions when they exist. Here’s how the process works for the given system:

• First, take the simpler of the two equations, solve for one variable, and express it in terms of the other. Here, we begin with \(2x - 7y = 8\) and solve for \(x\).
• Next, substitute the expression for \(x\) into the second equation \(-3x + \frac{21}{2}y = 5\). This allows us to eliminate one variable, narrowing it down to a single-variable linear equation.
• Upon substitution, simplify the resulting expression to see if the variable balances out or if a solution exists. If simplification produces an identity or contradiction, this clarifies the nature of the system (consistent, infinite solutions, or inconsistent, no solutions).

When done accurately, the substitution method provides reliable solutions or a clear indication of inconsistency or coincidence. It's essential to follow each substitution and simplification step closely to ensure accurate results.