Problem 5 - Algebra A Combined Function

Question

Determine whether each ordered pair is a solution of the system of linear equations. See Examples 1 and \(2 .\) \(\left\\{\begin{array}{l}2 y=4 x+6 \\ 2 x-y=-3\end{array}\right.\) a. (-3,-3) b. (0,3)

Step-by-Step Solution

Answer

(-3, -3) and (0, 3) both satisfy the system of equations, making them solutions.

Step 1: Introduction to the System of Equations

We are given a system of two equations: \(2y = 4x + 6\) and \(2x - y = -3\). We need to determine if the provided ordered pairs satisfy both equations.

Step 2: Evaluate Ordered Pair (-3, -3) - First Equation

Substitute \(x = -3\) and \(y = -3\) into the first equation: \(2y = 4x + 6\).Calculate: \(2(-3) = 4(-3) + 6\)\(-6 = -12 + 6\)\(-6 = -6\) - This pair satisfies the first equation.

Step 3: Evaluate Ordered Pair (-3, -3) - Second Equation

Now, substitute \(x = -3\) and \(y = -3\) into the second equation: \(2x - y = -3\).Calculate: \(2(-3) - (-3) = -3\)\(-6 + 3 = -3\)\(-3 = -3\) - This pair satisfies both equations.

Step 4: Evaluate Ordered Pair (0, 3) - First Equation

Now evaluate the second pair by substituting \(x = 0\) and \(y = 3\) into the first equation: \(2y = 4x + 6\).Calculate: \(2(3) = 4(0) + 6\)\(6 = 0 + 6\)\(6 = 6\) - This pair satisfies the first equation.

Step 5: Evaluate Ordered Pair (0, 3) - Second Equation

Substitute \(x = 0\) and \(y = 3\) into the second equation: \(2x - y = -3\).Calculate: \(2(0) - 3 = -3\)\(0 - 3 = -3\)\(-3 = -3\) - This pair satisfies both equations.

Key Concepts

Ordered Pairs

In the context of systems of linear equations, an "ordered pair" is a set of two numbers, often written in the form \(x, y\). The first component of the pair represents the value of \(x\) and the second represents the value of \(y\). Ordered pairs are essential for finding solutions to equations.
They give us the specific values for the variables that we need to check against the equations in the system.
Here’s how you can interpret ordered pairs:

• The pair \((-3, -3)\) implies \(x = -3\) and \(y = -3\).
• The pair \( (0, 3) \) implies \(x = 0\) and \(y = 3\).

To determine if an ordered pair is a solution to a system of equations, substitute the values of \(x\) and \(y\) into each equation and check if the equation holds true. If both equations are satisfied, the ordered pair is a solution to the system.

Substitution Method

The substitution method is a straightforward way to solve systems of equations, especially when checking if an ordered pair is a solution. In this method, you substitute the values from the ordered pair directly into the equations. Here’s the simple breakdown:

• Identify the variables in the system, \(x\) and \(y\) in most cases.
• Take each value from the ordered pair and substitute them into both given equations.
• Verify if the left side of each equation equals the right side. If they do, then the pair is a solution.

For example, substitute \(x = 0\) and \(y = 3\) into the equations \(2y = 4x + 6\) and \(2x - y = -3\), and evaluate whether they hold true.
The substitution method is particularly useful for checking potential solutions quickly and effectively, as you can see with the ordered pairs checked in this problem.

Solving Linear Equations

Solving linear equations is an essential skill when dealing with systems of equations. Each linear equation represents a straight line, and the solution gives points where the lines intersect.To solve a linear equation:

• Isolate one of the variables, usually starting with terms involving \(x\) or \(y\) on one side of the equation.
• Perform arithmetic operations such as addition, subtraction, multiplication, or division to simplify the equation.
• Check if a specific ordered pair satisfies the simplified equations by directly substituting the values.

In our given system, verification of the ordered pairs involves plugging in these values and checking if the expressions from both sides of the equations hold true. Skills in solving linear equations help ensure accuracy in such evaluations, allowing you to confirm solutions like \((-3, -3)\) or \( (0, 3) \) as valid based on the context of the system.