Problem 21 - Algebra 1: Concepts and Skills

Question

Determine whether the ordered pair is a solution of the equation. $$ -2 x-9 y=7,(-1,-1) $$

Step-by-Step Solution

Answer

The ordered pair (-1,-1) is not a solution to the equation -2x-9y=7.

Step 1: Insert the values

Insert the x and y values from the ordered pair into the equation. This will give us -2*(-1)-9*(-1).

Step 2: Simplify the equation

Simplify the equation. The equation becomes 2+9=11.

Step 3: Verify the result

Verify if the result on the left side equals to the result on the right side of the equation, which is 7. In this case, 11 is not equal to 7.

Key Concepts

Understanding Ordered Pairs

In mathematics, an ordered pair is a set of numbers used to represent a point in a coordinate system. The first number in the pair corresponds to the x-coordinate, and the second number corresponds to the y-coordinate. For example, in the ordered pair \((-1, -1)\), -1 is the x-coordinate, and -1 is the y-coordinate. Ordered pairs help us plot points on a graph and define the position of a point in a two-dimensional space. They are fundamental for working with equations in algebra, as they tell us where a graph intersects points along the x and y axes.
Remember:

• Ordered pairs are always written in parentheses.
• The sequence of numbers matters; \((a, b)\) is different from \((b, a)\).
• They are crucial in solving equations and finding solutions on a graph.

Using ordered pairs effectively means inserting the x and y values into equations to verify if they satisfy the given equation.

What are Linear Equations?

Linear equations are algebraic expressions that describe a straight line when graphed. These equations are called "linear" because they involve terms of the first degree, meaning they only include variables raised to the power of one. For example, an equation like \(-2x - 9y = 7\) is linear. Linear equations can have one or more variables, such as \(x\) and \(y\), and their general form is written as \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. Key Points:

• They model relationships between variables.
• The graph of a linear equation is always a straight line.
• These equations are used to find solutions that satisfy both terms, \(x\) and \(y\), simultaneously.

Understanding linear equations helps us to explore relationships between quantities and solve practical problems via algebra.

Verifying Solutions of Equations

Finding solutions to equations involves checking if specific ordered pairs make the equation true. For instance, when given the ordered pair \((-1, -1)\) and the equation \(-2x - 9y = 7\), we substitute \(x = -1\) and \(y = -1\) into the equation to see if both sides are equal. Let's go through the process:- Substitute values: \(-2(-1) - 9(-1)\).- Simplify both sides: \(2 + 9 = 11\).If the left side equals the right side of the equation, the ordered pair is a solution. In this exercise, since 11 does not equal 7, \((-1, -1)\) is not a solution.
Tips for verifying solutions:

• Carefully substitute the ordered pair values into the equation.
• Accurately perform arithmetic operations.
• Compare the simplified values to confirm if both sides are equal.

By following this method, you can confirm whether any ordered pair is a solution to the equation given.