Problem 37 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each ordered pair is a solution of the given equation. $$y=3 x \quad(2,3),(3,2),(-4,-12)$$

Step-by-Step Solution

Answer

Only the ordered pair (-4,-12) is a solution to the equation \(y = 3x\).

Step 1: Evaluate the first ordered pair (2,3)

Substitute the first ordered pair (2,3) into the equation. So \(x=2\) and \(y=3\). Check if \(y=3x\). This would mean checking if \(3=3*2\). But this is false, as 3 does not equal 6.

Step 2: Evaluate the second ordered pair (3,2)

Next, substitute \(x=3\) and \(y=2\) into the equation and check again. Thus, examine if \(2=3*3\). This also turns out to be false, as 2 does not equal 9.

Step 3: Evaluate the third ordered pair (-4,-12)

Finally, substitute \(x=-4\) and \(y=-12\) into the equation and check if \(y=3x\). In this case, it's necessary to check whether \(-12=3*(-4)\). Indeed, this is true. Thus, the pair (-4,-12) is a solution to the equation.

Key Concepts

Linear Equation

A linear equation is a mathematical expression that forms a straight line when plotted on a coordinate plane. It represents a relationship between two variables, typically denoted as \(x\) and \(y\). The general form of a linear equation in two variables is \(y = mx + c\), where \(m\) is the slope of the line and \(c\) is the y-intercept. This means for every unit increase in \(x\), \(y\) changes by \(m\) units.
In our example, the equation is \(y = 3x\), which is a specific case where the y-intercept \(c\) is zero. This means the line passes through the origin (0,0). The equation suggests that for every increase of 1 in \(x\), \(y\) increases by 3. Understanding these fundamental components helps in interpreting and solving linear equations effectively.

Solution Verification

Solution verification involves checking if a given ordered pair is a solution of a linear equation. An ordered pair is a set of numbers written in a specific order, often expressed as \((x, y)\). To verify if an ordered pair is a solution to the equation \(y = mx + c\), substitute \(x\) and \(y\) into the equation and check if both sides of the equation are equal.
For instance, let's consider the ordered pair \((2,3)\). By substituting \(x = 2\) and \(y = 3\) into \(y = 3x\), the equation becomes \(3 = 3 \times 2\), which simplifies to \(3 = 6\). Since both sides are not equal, \((2,3)\) is not a solution. Solution verification is straightforward and simply requires substitution and basic arithmetic operations to determine if a pair satisfies the equation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using a coordinate plane to solve geometric problems. This plane is defined by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). Each point in this plane is represented by an ordered pair \((x, y)\), which locates it in the two-dimensional space.
In our study, coordinate geometry allows us to visually assess where the solutions to the linear equation \(y = 3x\) lie on this plane. Each correct solution, such as the ordered pair \((-4,-12)\), lies exactly on the line described by the equation. Ordered pairs that do not lie on the line indicate they are not solutions to the equation. Understanding coordinate geometry is crucial for accurately interpreting and manipulating linear equations and their graphical representations.