Problem 60 - College Algebra

Question

Which one of the following is true? a. If the coordinates of a point satisfy the inequality \(x y>0,\) then \((x, y)\) must be in quadrant I. b. The ordered pair \((2,5)\) satisfies \(3 y-2 x=-4\) c. If a point is on the \(x\) -axis, it is neither up nor down, so \(x=0\) d. None of the above is true.

Step-by-Step Solution

Answer

The correct choice is d. None of the above is true.

Step 1: Check Statement a

Given an inequality \(x y>0\). This inequality states that the product of the coordinates \(x\) and \(y\) are greater than zero. In a coordinate plane, this is possible in both first quadrant (where \(x > 0\) and \(y > 0\)) and third quadrant (where \(x < 0\) and \(y < 0\)). Therefore, the statement 'If the coordinates of a point satisfy the inequality \(x y>0\), then \((x, y)\) must be in quadrant I' is not true, as it rejects the possibility of the point being in the third quadrant.

Step 2: Check Statement b

The statement gives us an ordered pair \((2,5)\) which allegedly satisfies the given equation; so we substitute the values into the equation to verify. The equation is \(3y - 2x = -4\), substituting \(x=2\) and \(y=5\) yields \(3(5) - 2(2) = 11\), which is not equal to -4. So, this statement is not true.

Step 3: Check Statement c

A point on the x-axis will always have a y-coordinate of zero, since it is neither up nor down from the x-axis. The x-coordinate can be any real number. So, the given statement 'If a point is on the x-axis, it is neither up nor down, so \(x=0\)' is not true as it should be \(y=0\)

Step 4: Check Statement d

The previous steps have shown that statement a, b, and c are all false, which means the answer should be d. 'None of the above is true'.

Key Concepts

Quadrants of Coordinate Plane

The coordinate plane is divided into four sections, each called a quadrant. These are numbered counterclockwise starting from the top right. In Quadrant I, both x and y values are positive, while in Quadrant II, x is negative and y is positive. In Quadrant III, both are negative, and finally, in Quadrant IV, x is positive and y is negative.

This understanding is crucial when solving inequalities such as the exercise statement a, which incorrectly assumes that an inequality with a positive product, specifically of the form xy > 0, must pertain to only Quadrant I. However, a product can also be positive if both numbers are negative, which applies to Quadrant III as well. Thus, the inequality xy > 0 could represent points situated in either Quadrant I or III.

Ordered Pairs

Ordered pairs, denoted as (x, y), represent the position of points on a coordinate plane. 'x' corresponds to the horizontal component while 'y' represents the vertical component. For instance, the ordered pair (2, 5) from the exercise implies the point is located 2 units right and 5 units up from the origin.

Verifying if an ordered pair satisfies an equation is straightforward; it only involves substituting 'x' and 'y' into the equation and checking if the equality holds. In the exercise's Statement b, the substitution of (2, 5) into the equation 3y - 2x = -4 doesn't result in a true equality, hence (2, 5) does not satisfy the equation.

Solving Linear Equations

Solving linear equations involves finding the values of the variables that make the equation true. Equations on a coordinate plane often depict a line, and the coordinates that lie on that line will satisfy the equation.

In the context of the coordinate plane, a line may intersect the x-axis or y-axis. A common misconception, as seen in the exercise's Statement c, is confusing the x-value of points on the y-axis with those on the x-axis. It is important to note that for any point on the y-axis, the x-value is always zero because it's directly above or below the origin with no horizontal deviation. Conversely, a point on the x-axis will have a y-value of zero. The correction of such misunderstandings is integral to solving linear equations accurately.