Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Solve each system by substitution. $$\begin{aligned}&8 x+3 y=2\\\&5 x=17+6 y\end{aligned}$$

Step-by-Step Solution

Answer

The solution is \((x, y) = (1, -2)\).

Step 1: Solve One Equation for One Variable

Let's solve the second equation for one variable, say, for \(x\). The equation is \(5x = 17 + 6y\). We can rewrite it to express \(x\) in terms of \(y\): \(x = \frac{17 + 6y}{5}\).

Step 2: Substitute Back into the Other Equation

Now, we substitute \(x = \frac{17 + 6y}{5}\) into the first equation \(8x + 3y = 2\). This gives us:\[8\left(\frac{17 + 6y}{5}\right) + 3y = 2\]

Step 3: Simplify the Substitution

Simplify the expression from Step 2:\[\frac{8(17 + 6y)}{5} + 3y = 2\]Multiply the equation by 5 to eliminate the fraction:\[8(17 + 6y) + 15y = 10\]Distribute the terms:\[136 + 48y + 15y = 10\]

Step 4: Solve for \(y\)

Combine like terms:\[63y = 10 - 136\]which simplifies to:\[63y = -126\]From this, solve for \(y\):\(y = \frac{-126}{63} = -2\)

Step 5: Substitute Back to Find \(x\)

Substitute \(y = -2\) back into the expression for \(x\):\[x = \frac{17 + 6(-2)}{5}\]This simplifies to:\[x = \frac{17 - 12}{5} = \frac{5}{5} = 1\]

Step 6: Verify the Solution

Check that \((x, y) = (1, -2)\) satisfies both original equations:1. For \(8x + 3y = 2\): \[8(1) + 3(-2) = 8 - 6 = 2\] (True)2. For \(5x = 17 + 6y\): \[5(1) = 17 + 6(-2) = 5 = 17 - 12\] (True)The solution satisfies both equations.

Key Concepts

Substitution Method

The substitution method is a handy way to solve systems of linear equations. It's particularly useful when you can easily solve one of the equations for one variable. Here's a quick roundup on how it generally works:

• First, solve one of the equations for one of the variables. This involves isolating the variable on one side of the equation. For example, if you have an equation like \(5x = 17 + 6y\), you can rearrange it to solve for \(x\), resulting in \(x = \frac{17 + 6y}{5}\).
• Next, substitute this expression into the other equation wherever the isolated variable appears. This replaces one variable in the equation, helping to reduce the number of variables.
• After substituting, you end up with an equation that has just one variable, which simplifies the entire process of finding the solution.

It's a great method for maintaining control over the calculations and simplifying complex-looking problems into manageable tasks.

Linear Equations

Linear equations are algebraic equations where the highest exponent of any variable is one. They're called 'linear' because they graph as straight lines on a coordinate plane.When dealing with a system of linear equations, you're finding a common solution or set of values for \(x\) and \(y\) that satisfies each equation in the system simultaneously.Some important points about linear equations:

• They have variables (like \(x\) and \(y\)) but no variable is raised to a power higher than one.
• They can often represent real-world scenarios, like calculating costs, distances, or other relations.
• In a graph, they appear as straight lines, and the point where the lines intersect represents the solution to the system.

In the context of solving a system of equations, understanding that each equation graphically represents a line helps you visually interpret the solution you derive algebraically.

Equation Solving

Equation solving is the process of finding the value of the unknown variables that make the equation true. In the case of a system of equations, you're working to find a set of values that satisfy all the equations in the system at once. Here’s a basic way to approach equation solving:

• Identify the variables and equations in the problem.
• Use methods like substitution or elimination to reduce the number of variables, simplifying the equations as needed.
• Perform algebraic manipulations to isolate the variable on one side of the equation.
• Check your solution by substituting the values back into the original equation to ensure it satisfies all equations in the system.

This systematic approach ensures that no steps are skipped and the solution is both accurate and complete. Always double-check your solutions to verify they hold true in the original contextual framework.