Problem 61 - Intermediate Algebra

Question

Decide whether the ordered pair \((-4,3)\) is a solution of the system of linear equations. $$ \left\\{\begin{array}{l} 4 x-y=-19 \\ 3 x+2 y=-6 \end{array}\right. $$

Step-by-Step Solution

Answer

Yes, \((-4, 3)\) is a solution to the system of equations.

Step 1: Substitute into First Equation

Substitute the values of the ordered pair \((-4, 3)\) into the first equation: \(4x - y = -19\). This gives us: \(4(-4) - 3 = -19\). Simplify the left side to get \(-16 - 3 = -19\). Hence, \(-19 = -19\) which means the first equation is satisfied.

Step 2: Substitute into Second Equation

Substitute the values of the ordered pair \((-4, 3)\) into the second equation: \(3x + 2y = -6\). This gives us: \(3(-4) + 2(3) = -6\). Simplify the left side to get \(-12 + 6 = -6\). Hence, \(-6 = -6\) which means the second equation is also satisfied.

Step 3: Verify Solution

Since both equations are satisfied, the ordered pair \((-4, 3)\) is a solution to the system of equations.

Key Concepts

Understanding Ordered Pairs

An ordered pair is a fundamental concept used to represent a specific point on a plane. It consists of two elements: the first is the x-coordinate and the second is the y-coordinate. In our exercise, the ordered pair is \((-4, 3)\).

• The x-coordinate is -4, which tells us how far left from the origin (0,0) the point is located.
• The y-coordinate is 3, indicating how far above the x-axis the point is.

These pairs play a crucial role in defining solutions for systems of equations, as they tell us the specific values of variables that satisfy the given equations. Remember, for ordered pairs to be considered a solution, they must satisfy all equations in the system simultaneously.

Substituting Values in Equations

Substitution is a method used to determine whether an ordered pair is a solution to a system of equations. This involves replacing the variables in the equation with the values from the given ordered pair.
In our exercise, let's break down the substitution process:

1. **First Equation:** Replace \(x\) with -4 and \(y\) with 3 in \(4x - y = -19\).

• Calculate \(4(-4) - 3\).
• Simplify to get \(-19\), satisfying the first equation since \(-19 = -19\).

2. **Second Equation:** Substitute the same \(x\) and \(y\) values into the second equation, \(3x + 2y = -6\).

• Calculate \(3(-4) + 2(3)\).
• Simplify to get \(-6\), matching the right side of the equation \(-6 = -6\).

By substituting these values, you can verify if both conditions of the system are fulfilled.

Solution Verification

Verifying a solution is a vital step in confirming whether the proposed ordered pair actually solves the system of equations. After substituting the values and performing the calculations, we need to ensure that both equations are satisfied.
Here's how you verify:

• Check if the result from substituting the ordered pair into the first equation equals the constant on the right side. In this case, verify that \(-19\) is indeed equal to \(-19\).
• For the second equation, confirm that the calculated left side equals \(-6\), which it does.

Since both sides of each equation match their respective constants after substitution, the ordered pair is indeed a valid solution. Verification ensures that no errors occurred during substitution and calculation steps, further confirming the correctness of the solution.