Problem 54 - Understanding Elementary Algebra with Geometry

Question

Given the equation \(4 x-y=8,\) complete the given ordered pairs: $$(-2, \quad) \quad(0, \quad) \quad(\quad, 4) \quad(\quad, 0)$$

Step-by-Step Solution

Answer

(-2, -16), (0, -8), (3, 4), (2, 0).

Step 1: Solve for y when x = -2

Substitute x = -2 into the equation 4x - y = 8. This gives: 4(-2) - y = 8 which simplifies to -8 - y = 8. Solve for y: -y = 8 + 8, so -y = 16. Therefore, y = -16. The ordered pair is: (-2, -16).

Step 2: Solve for y when x = 0

Substitute x = 0 into the equation 4x - y = 8. This gives: 4(0) - y = 8 which simplifies to 0 - y = 8. Solve for y: -y = 8, so y = -8. The ordered pair is: (0, -8).

Step 3: Solve for x when y = 4

Substitute y = 4 into the equation 4x - y = 8. This gives: 4x - 4 = 8. Add 4 to both sides: 4x = 8 + 4, which simplifies to 4x = 12. Divide by 4: x = 12 / 4, so x = 3. The ordered pair is: (3, 4).

Step 4: Solve for x when y = 0

Substitute y = 0 into the equation 4x - y = 8. This gives: 4x - 0 = 8 which simplifies to 4x = 8. Divide by 4: x = 8 / 4, so x = 2. The ordered pair is: (2, 0).

Key Concepts

ordered pairs

In mathematics, an ordered pair is a pair of elements written in a specific order. Typically, it is presented as \((x, y)\) in the Cartesian coordinate system where \(x\) is the first element and \(y\) is the second. Ordered pairs are essential when working with equations involving two variables. The position of each element in the pair is crucial because it defines a point in a two-dimensional space.
Consider the equation given in the exercise: \(4x - y = 8\). To find the corresponding ordered pairs, you substitute specific values for one variable and solve for the other. For instance:
\(\bullet\) To find the pair \((-2, y)\), substitute \(-2\) for \(x\) and solve for \(y\).
\(\bullet\) For the pair \(x, 4)\), substitute 4 for \(y\) and solve for \(x\).
The resulting values form the ordered pairs that satisfy the given equation.

substitution method

The substitution method is a technique used to solve systems of equations by substituting one equation into another. Here, we use this method to find the values of \(x\) or \(y\) that satisfy the equation \(4x - y = 8\).

Step-by-Step Process:
1. Identify which variable to solve for first. For example, when given \(x = -2 \) substitute it in the equation: \( 4(-2) - y = 8 \)
2. Simplify the left side: -8 - y = 8
3. Isolate the variable by performing algebraic operations: - y = 16 \ \rightarrow y = -16
This tells us one ordered pair is (-2, -16).
Continue using the substitution method for other pairs by substituting given values and solving for the unknown variable.

linear equations in two variables

Linear equations in two variables take the form \(ax + by = c\). These equations graph as straight lines in the Cartesian plane.

For our given equation \(4x - y = 8\):

• The coefficients of \(x \) and \(y \) (4 and -1, respectively) dictate the slope and intercepts.
• Simplifying or solving for one variable at a time reveals the relationship between \( x \) and \( y \).
• Solving this particular equation using the substitution method helps us find points on the line represented by specific ordered pairs.

By substituting values, both variables’ values can be calculated, and we obtain specific ordered pairs that lie on the line of the given linear equation.