Problem 3 - Algebra A Combined Function

Question

Solve each system of equations by the substitution method. $$ \left\\{\begin{array}{l} x+y=6 \\ y=-3 x \end{array}\right. $$

Step-by-Step Solution

Answer

The solution to the system is \( x = -3 \) and \( y = 9 \).

Step 1: Solve One Equation for One Variable

The system of equations is: \( x+y=6 \) and \( y=-3x \) . We notice that the second equation \( y = -3x \) already gives \( y \) in terms of \( x \).

Step 2: Substitute the Expression into the Other Equation

Substitute \( y = -3x \) from the second equation into the first equation \( x + y = 6 \). This gives us \( x + (-3x) = 6 \).

Step 3: Simplify and Solve for One Variable

Simplify the substituted equation: \( x - 3x = 6 \), which simplifies to \( -2x = 6 \). Divide both sides by \(-2\) to solve for \( x \): \( x = -3 \).

Step 4: Substitute Back to Find the Other Variable

Now that we have \( x = -3 \), substitute it back into \( y = -3x \) to find \( y \). This gives us \( y = -3(-3) = 9 \).

Step 5: Verify the Solution

Check the solution \((x, y) = (-3, 9)\) by plugging \( x = -3 \) and \( y = 9 \) back into the original equations: \( -3 + 9 = 6 \) which holds true, and \( 9 = -3(-3) \) which also holds true.

Key Concepts

System of Equations

A system of equations consists of two or more equations with the same set of variables. These equations need to be solved together to find a common solution. In our original exercise, we have a system of two equations:

• Equation 1: \( x+y=6 \)
• Equation 2: \( y=-3x \)

These equations share the same variables, \(x\) and \(y\). The objective is to find values for these variables that satisfy both equations simultaneously. Often, systems of equations represent real-world scenarios where multiple conditions must be true at the same time. By solving these systems, we can determine the necessary conditions that will satisfy the given relationships.

Solve Equations

Solving equations is the process of determining the values of variables that make the equations true. For solving the system of equations using the substitution method, we follow a systematic approach:1. **Understand the equations**: First, we recognize that equation 2, \( y = -3x \), already provides us with \(y\) expressed in terms of \(x\).
2. **Substitute known values**: Next, we substitute this expression for \(y\) into equation 1, \( x+y=6 \), allowing us to solve for \(x\) independently.
3. **Solve for one variable**: After substitution, we simplify and solve the remaining equation to find the value of \(x\).
4. **Find the other variable**: Finally, we revisit the substituted equation to solve for \(y\) using the value of \(x\) obtained in the earlier steps.This structured process helps in isolating variables, simplifying the task of solving complex systems.

Algebraic Manipulation

Algebraic manipulation involves performing operations to transform equations into a more tractable form. This is crucial when using methods like substitution. In our solution:

• After substituting \(y = -3x\) into \(x + y = 6\), we perform operations to simplify the equation into \(-2x = 6\).
• Dividing both sides by \(-2\) allows us to solve for \(x\), yielding \(x = -3\).

These manipulations include:- **Substitution**: Replacing one variable with an equivalent expression from another equation.- **Simplification**: Combining like terms to simplify the equation.- **Isolation**: Solving for a specific variable by isolating it on one side of the equation.Each algebraic step requires careful attention to ensure that the logical and mathematical integrity of the equation is maintained.

Verification of Solution

Verification of the solution is an essential step to ensure the accuracy of the values obtained. This involves substituting the values of \(x\) and \(y\) back into the original equations to check if they hold true:

• Substituting \((x, y) = (-3, 9)\) into Equation 1: \( -3 + 9 = 6 \) confirms that the left side equals the right side.
• Similarly, substituting into Equation 2: \( 9 = -3(-3) \) also verifies each side of the equation is equivalent.

Verification helps in:- **Confirming accuracy**: Ensures that the solution satisfies all given equations.- **Building confidence**: Reinforces that the solution obtained is the correct one.- **Identifying errors**: If any discrepancies arise, it indicates a need to revisit the problem-solving steps for errors.