Problem 15 - Algebra 1: Concepts and Skills

Question

Determine whether the given ordered pair is a solution of the equation. $$ y=3,(3,-3) $$

Step-by-Step Solution

Answer

No, the ordered pair (3, -3) is not a solution to the equation \( y = 3 \).

Step 1: Understand the equation and ordered pair

The equation is \( y = 3 \) and the ordered pair given is \( (3, -3) \). In the pair, 3, is the x-coordinate and -3 is the y-coordinate.i.e \( x = 3 \) and \( y = -3 \).

Step 2: Check if equation and y-coordinate of ordered pair are equal

For the ordered pair to be a solution to the equation, the y-value in the equation and the y-coordinate in the ordered pair must be equal. So, this would entail comparing the y-value of the equation, which is 3 with that of the ordered pair, which is -3.

Step 3: Evaluate the Result

By comparing the values in step 2, it can be established that the y-value in the equation (3) is not same as the y-coordinate in the ordered pair (-3). Thus, the given ordered pair is not a solution of the equation.

Key Concepts

Understanding Linear Equations

Linear equations are equations that create straight lines when plotted on a graph. A basic form of a linear equation involves one or more variables, usually expressed as "y = mx + b" where "m" is the slope and "b" is the y-intercept. However, in simpler terms, as seen in our exercise, it can also be as straightforward as "y = a constant value".

This particular type of equation means that for every x-value, the y-value remains unchanged at the given constant. This creates a horizontal line across the graph for its solutions.

• Slope: In the equation form "y = mx + b", "m" is the slope that indicates the steepness and direction of the line.
• Constant y-value: When y is equal to a constant number, it forms a specific horizontal line in the coordinate plane.

Recognizing these forms helps in quickly identifying the nature of the graph and its solutions. The focus is on maintaining a straight path with consistent values dependent on the type of linear equation presented.

Understanding Solutions in Equations

In mathematics, a solution to an equation is a value or a set of values that satisfy the equation. For linear equations, solutions are often expressed as ordered pairs (x, y), where substituting these values into the equation makes the equation true.

To determine if an ordered pair is a solution, substitute the x and y coordinates into the equation and verify if the left-hand side equals the right-hand side. In our exercise, the equation "y = 3" needs the ordered pair's y-coordinate to be 3 for it to be a solution.

• Ordered Pair: Represents the x and y coordinates, in the form (x, y).
• Substitution: Replacing variables with numbers from the ordered pair to see if they satisfy the equation.
• Verification: Ensuring that both sides of the equation match after substitution.

When checking for solutions, attention to detail is important. Even a small mistake in substitution or verification can lead to incorrect conclusions about the solution.

Plotting on a Coordinate Plane

The coordinate plane is a two-dimensional plane defined by the x-axis (horizontal) and y-axis (vertical) intersecting at a point called the origin (0,0). This system is used to graphically display solutions to equations as points or lines.

Each point on the coordinate plane is represented by an ordered pair, indicating its position relative to the x and y axes. One can plot the solutions of linear equations on this plane to visualize them better.

• X-axis and Y-axis: The two perpendicular lines that define the coordinate plane where values are plotted.
• Origin: The point (0,0) where the x-axis and y-axis meet.
• Graphical Representation: Helps in visualizing the relationship between variables in an equation.

By plotting points, you can observe patterns, such as the linear nature of solutions to linear equations. This practice not only aids in understanding but also in verifying solutions through visual means.