Problem 123 - College Algebra

Question

Find the ordered pairs (______, 0) and (0,______) satisfying 4 x-3 y-6=0

Step-by-Step Solution

Answer

The ordered pairs are \((1.5, 0)\) and \((0, -2)\)

Step 1: Replace y with 0 and Solve for x

First, we can find the x-value when y=0. We substitute y=0 into the equation \(4x - 3*0 - 6 = 0\). After simplifying, this becomes \(4x = 6\). To isolate x, we can divide through by 4 which gives us \(x = 1.5\) or \(x = \frac{3}{2}\) in more simple terms.

Step 2: Replace x with 0 and Solve for y

Next, to find the y-value when x=0, we substitute x=0 into the main equation, we have \(4*0 - 3y - 6 = 0\). Simplifying the equation we get \(-3y = 6\).Solving for y, we divide through by -3 gives us \(y = -2\).

Step 3: Identify the ordered pairs

Based on the solution, the ordered pairs are \((1.5, 0)\) or in simple terms \((\frac{3}{2},0)\) when y=0 and \((0, -2)\) when x=0.

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in coordinate geometry, representing a system that can pinpoint an exact location on a plane using two numbers. Usually, an ordered pair is written in the form \( (x, y) \) where \( x \) represents the horizontal component, or 'x-coordinate', and \( y \) represents the vertical component, or 'y-coordinate'.

In the context of algebra, finding ordered pairs often involves solving equations to ascertain the values of \( x \) and \( y \) that satisfy the equation. An essential part of solving for ordered pairs is understanding that each pair corresponds to a point on a graph. For example, in our exercise, the ordered pairs that satisfy the equation \( 4x - 3y - 6=0 \) are \( (1.5, 0) \) and \( (0, -2) \) which denote points on the cartesian plane. The first number in each pair is found by setting \( y \) to zero and solving for \( x \) which gives us the intersection with the \( x \) axis, while the second number is found by setting \( x \) to zero and finding \( y \) to get the point where the line meets the \( y \) axis.

Algebraic Solutions

Algebraic solutions refer to the process of finding the values for unknowns in an equation by manipulating the equation until the unknowns are isolated. This method involves a variety of techniques such as adding, subtracting, multiplying, and dividing, which allows us to simplify the equation and solve for the variables.

In our exercise, algebraic manipulation was used to solve for \( x \) and \( y \) in a linear equation. When we were given \( y = 0 \) to find the corresponding \( x \) value, we performed algebraic operations to isolate \( x \) and found it to be \( \frac{3}{2} \). Similarly, to find the corresponding \( y \) value when \( x = 0 \) was given, we isolated \( y \) and found it to be \( -2 \). It is vital in algebra not only to perform operations accurately but also to check that the solution fits the original equation, ensuring the solution is correct.

Substitution Method

The substitution method is a powerful tool in algebra used to solve systems of equations. It involves replacing a variable with an equivalent expression obtained from another equation, making it possible to solve for one variable at a time. It often simplifies an equation or a system of equations so that it becomes easier to solve.

Applying the substitution method in our exercise helped us to determine the values of the ordered pairs. By substituting \( 0 \) for \( y \) in the given equation, we reduced the equation down to \( 4x - 6 = 0 \), making it straightforward to solve for \( x \). Similarly, replacing \( x \) with \( 0 \) led us to the equation \( -3y - 6 = 0 \), from which we could solve for \( y \). This method is not just limited to single equations like in our exercise. It's widely used in solving simultaneous equations, which is where its true utility shines as it allows us to find the point of intersection between two lines graphically represented by the equations.