Problem 38 - Introductory Algebra for College Students

Question

determine whether each ordered pair is a solution of the given equation. $$y=4 x \quad(3,12),(12,3),(-5,-20)$$

Step-by-Step Solution

Answer

The ordered pairs that are solutions to the equation \(y=4x\) are (3,12) and (-5,-20). The pair (12,3) is not a solution.

Step 1: Substitute the first ordered pair

Start by substituting the values of the first ordered pair into the equation. For the pair (3,12), x=3 and y=12. When these values are substituted in, the equation becomes \(12=4*3\) or \(12=12\). Since the result is true, the pair (3,12) satisfies the equation.

Step 2: Substitute the second ordered pair

Next, substitute the values of the second ordered pair into the equation. For the pair (12,3), x=12 and y=3. Substituting these values gives us \(3=4*12\) or \(3=48\). This is not true, so the pair (12,3) is not a solution to the equation.

Step 3: Substitute the third ordered pair

Finally, substitute the values of the third ordered pair into the equation. For the pair (-5,-20), x=-5 and y=-20. Substituting these values gives us \(-20=4*(-5)\) or \(-20=-20\). Since the result is true, the pair (-5,-20) is a solution to the equation.

Key Concepts

Ordered Pairs

An ordered pair is simply a pair of numbers used to locate a point on a grid, often referred to as the XY plane. This pair consists of an input and an output, usually denoted as

• x: the first number in the pair, representing the horizontal position on the x-axis.
• y: the second number, representing the vertical position on the y-axis.

The notation for ordered pairs looks like this: (x, y).
Understanding how ordered pairs work is crucial when dealing with equations, as each pair can be a potential solution. To determine this, each x value from a pair is substituted into an equation to see if the corresponding y value results logically.
In the context of our exercise, we're checking if certain ordered pairs are solutions to the equation \(y = 4x\). This involves substituting the x and y from each ordered pair into the equation to check if the equation holds true.

Substitution Method

The substitution method is an effective way to determine whether an ordered pair satisfies a given linear equation. The process involves replacing a variable in the equation with its corresponding numerical value from the ordered pair.
Here's how you can apply the substitution method step-by-step:

• Take the ordered pair, which consists of an x and a y value.
• Substitute the x value into the equation wherever x appears.
• Substitute the y value into the equation wherever y appears.

If after substitution, the left side of the equation equals the right side, then the ordered pair is a solution to the equation.
In our exercise, when substituting each pair into \(y = 4x\), only pairs like (3,12) and (-5,-20) satisfy the equation, as they make both sides equal.

Solution Verification

Solution verification is the process of checking whether the substitutions made in an equation are correct and whether they satisfy the original equation. This step helps confirm that the chosen ordered pairs are indeed solutions to the given equation.
To verify a solution:

• Use the substituted values and simplify each side of the equation.
• Check if both sides are equal. If they are, the ordered pair is a valid solution.
• If they are not equal, the pair is not a solution.

In our earlier examples, we verified that (3,12) and (-5,-20) are solutions since their substitutions made both sides of the equation equal, aligning perfectly with \(y = 4x\).
This method is invaluable for ensuring accuracy, especially when working with more complex equations, because it offers a concrete way to affirm the correctness of a solution.