Problem 19 - Algebra 1

Question

Decide whether the given ordered pair is a solution of the equation. \(6 y-3 x=-9,(2,-1)\)

Step-by-Step Solution

Answer

The ordered pair \((2,-1)\) is not a solution to the equation \(6y - 3x = -9\).

Step 1: Identify the given

In this problem, the given ordered pair is \((2,-1)\) which means \(x = 2\) and \(y = -1\). And the equation given is \(6y - 3x = -9\)

Step 2: Substitute the ordered pair into the equation

Next, we will replace the variables \(x\) and \(y\) in the equation with the values from the ordered pair, \(x = 2\) and \(y = -1\). On doing so, the equation becomes \(6*(-1) - 3*2 = -9\)

Step 3: Simplify the equation

Now, simplifying the equation we get \(-6 - 6 = -12\)

Step 4: Compare the simplified equation with the given equation

Since the results of both sides of the equation are not equal, it means that \((2,-1)\) is not a solution to the equation \(6y - 3x = -9\)

Key Concepts

Ordered Pairs

When working with algebraic equations, an important concept is that of ordered pairs. An ordered pair consists of two elements, usually written as \((x, y)\), where the order of these elements matters. The first element represents the value for the variable \(x\) and the second element represents the value for the variable \(y\). Ordered pairs are often used to specify coordinates on a graph, but they play a crucial role in resolving many algebraic problems as well.
In our exercise, the ordered pair \((2, -1)\) is examined to see if it solves the equation \(6y - 3x = -9\). This means that we will substitute \(x = 2\) and \(y = -1\) into the equation to verify if it satisfies the equation's requirement. Understanding how ordered pairs work and how to use them in equations is essential in algebra.

Substitution Method

The substitution method is a key technique in solving algebraic equations, especially when you want to check if a particular solution works for an equation. This method involves replacing variables in an equation with specific numbers to see if the equation holds true. In our case, we substitute the values from the ordered pair \((x, y)\) into the equation.

• First, identify the values for each variable from the ordered pair. For example, in \((2, -1)\), \(x = 2\) and \(y = -1\).
• Next, replace every occurrence of \(x\) and \(y\) in the equation \(6y - 3x = -9\) with these numbers.

After substitution, the equation becomes \(6(-1) - 3(2) = -9\). Solving this will help us determine if the ordered pair is a solution.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. They generally contain variables, numbers, and operations. One of the main goals when dealing with algebraic equations is to find values for the variables that make the equation true.
In the problem considered here, the equation is \(6y - 3x = -9\). To check if the ordered pair \((2, -1)\) satisfies this equation, we slide into the realm of algebra by substituting \(x = 2\) and \(y = -1\).
Once substituted, the goal is to simplify the equation. In this case, simplifying yields \(-6 - 6 = -12\). Since \(-12\) does not equal \(-9\), the equation is not satisfied by the ordered pair \((2, -1)\). This example shows how algebraic equations use variables to represent numbers that can change, and how solving them involves finding the right values to satisfy the equation.