Problem 8 - Precalculus: Functions and Graphs

Question

Use the method of substitution to solve the system. $$\left\\{\begin{array}{l} 3 x-4 y+20=0 \\ 3 x+2 y+8=0 \end{array}\right.$$

Step-by-Step Solution

Answer

The solution to the system is \(x = -4\) and \(y = 2\).

Step 1: Solve for x

First, we'll solve one of the equations for one of the variables. Let's take the first equation: \(3x - 4y + 20 = 0\) and solve for \(x\). We can rearrange it by moving terms around:\[3x = 4y - 20\]Now, divide both sides by 3:\[x = \frac{4y - 20}{3}\]

Step 2: Substitute x in the Second Equation

Now, take the expression we found for \(x\) in Step 1 and substitute it into the second equation. The second equation is \(3x + 2y + 8 = 0\). Substitute the expression for \(x\):\[3\left(\frac{4y - 20}{3}\right) + 2y + 8 = 0\]Simplify by multiplying:\[4y - 20 + 2y + 8 = 0\]

Step 3: Solve for y

Now, let's solve for \(y\). Simplify the equation from Step 2:\[6y - 12 = 0\]Add 12 to both sides:\[6y = 12\]Divide both sides by 6:\[y = 2\]

Step 4: Substitute y back into the Expression for x

With \(y\) found, we can substitute \(y = 2\) back into the expression for \(x\) found in Step 1:\[x = \frac{4(2) - 20}{3}\]Calculate:\[x = \frac{8 - 20}{3} = \frac{-12}{3} = -4\]

Step 5: Double Check the Solution

It's important to verify the solution by substituting the found values back into the original equations. For the first equation:\[3(-4) - 4(2) + 20 = -12 - 8 + 20 = 0\]For the second equation:\[3(-4) + 2(2) + 8 = -12 + 4 + 8 = 0\]Both equations hold true, confirming our solution.

Key Concepts

Method of Substitution

The method of substitution is a technique used to solve a system of linear equations. It involves breaking down the problem into smaller, more manageable steps.
This method is particularly useful when one equation is relatively easy to solve for a single variable.

Let's take a closer look at the approach:

• First, you solve one of the equations for one of the variables. In simpler terms, rearrange the equation so that one variable is isolated. This means expressing one variable in terms of the other.
• With this equation, substitute the expression you just found into the other equation. This means wherever you see that variable in the other equation, replace it with the expression.
• You're then left with one equation with one unknown variable, making it easier to solve.

By using substitution, you are systematically reducing a two-variable problem down to a one-variable problem. This reduction greatly simplifies the task of finding the solution. In the context of the given exercise, the substitution method was used to find expressions for variables effectively, paving the way for finding exact values.

Solving for Variables

Once you have used the substitution method to express one variable in terms of another, the next step is solving for the variable.
This involves basic algebraic manipulations to isolate the variable.

Here’s a step-by-step breakdown:

• Substitute the expression from the first step into the second equation. This is crucial because it reduces the system of equations to a single equation with only one variable.
• Perform algebraic operations to simplify the equation further. Start by combining like terms and simplifying the expressions as much as possible.
• Continue by isolating the variable on one side of the equation, ensuring all terms involving the variable are on one side, and constants on the other.
• Finally, solve for the variable by performing inverse operations, such as dividing by the coefficient of the variable, to solve it directly.

For our system, we solved for \(y\) first after substituting. Then, using this value, we substituted back to find \(x\). Each variable is solved methodically by maintaining balance in the equations.

Verification of Solution

Verifying your solution is an essential final step in solving systems of equations.
This ensures that the values found truly satisfy the original equations.Verification acts as a double-check to prevent errors in your calculations.

Here is how you can verify your solution:

• Take the values found for each variable and substitute them back into the original equations.
• Calculate each side of the equations to check if both sides are equal for each equation.
• If both equations hold true, this confirms that your solution is correct.

For example, in our exercise, once we found \(x = -4\) and \(y = 2\), we substituted these values back into both original equations. Both checks held true, which confirmed the accuracy of our solution. Ensuring correctness gives confidence that the solution is reliable and robust.