Problem 29 - Algebra 1

Question

Find three different ordered pairs that are solutions of the equation. \(y=4\left(\frac{1}{2} x-1\right)\)

Step-by-Step Solution

Answer

The three different ordered pairs that are solutions of the equation are (0, -4), (2, 0), and (4, 4).

Step 1: Substitute x = 0

Let's first substitute x = 0 into the equation. This gives \(y = 4\left(\frac{1}{2}\cdot 0 - 1\right) = 4\left(-1\right) = -4\). Therefore, one solution is (0, -4).

Step 2: Substitute x = 2

Next, substitute x = 2 into the equation. This gives \(y = 4\left(\frac{1}{2}\cdot 2 - 1\right) = 4\cdot 0 = 0\). Thus, another solution is (2, 0).

Step 3: Substitute x = 4

Lastly, substitute x = 4 into the equation. This yields \(y = 4\left(\frac{1}{2}\cdot 4 - 1\right) = 4\cdot 1 = 4\). This gives us a third solution, (4, 4).

Key Concepts

Ordered Pairs

Understanding what ordered pairs are is fundamental in solving linear equations. An ordered pair is a pair of numbers written in a specific order, usually as \( (x, y) \), where \( x \) represents the horizontal position on a graph and \( y \) represents the vertical position. The first value \( x \) is called the 'abscissa' and the second value \( y \) is referred to as the 'ordinate'.

When we say we are finding ordered pairs that are solutions of the equation \( y = 4(\frac{1}{2} x - 1) \), we are looking for pairs of \( x \) and \( y \) values that make the equation true when substituted in. These pairs can be plotted on a coordinate grid, resulting in a visual representation of the solutions. Plotting multiple pairs that satisfy the equation gives us a line, showing every possible pair that is a solution to the equation. This is crucial because it helps us visualize the relationship between \( x \) and \( y \) established by the equation.

Substitution Method

The substitution method is a technique used to find the solutions of a function or system of equations. It involves replacing variables with specific values to solve for other variables. For example, if we have an equation \( y = 4(\frac{1}{2} x - 1) \), we can solve for \( y \) by substituting different values of \( x \) into the equation.

The process starts with choosing a value for \( x \) and 'substituting' it into the equation to find the corresponding \( y \) value. This is exactly what was done in the given problem: substituting \( x = 0 \) gave the ordered pair \( (0, -4) \), substituting \( x = 2 \) gave \( (2, 0) \) and substituting \( x = 4 \) gave \( (4, 4) \). These steps not only provide specific solutions but also illustrate how changing \( x \) affects \( y \) in a linear relationship.

Linear Function

A linear function is an algebraic equation involving two variables, typically \( x \) and \( y \) that produces a straight line when graphed on a coordinate plane. The general form of a linear function is \( y = mx + b \) where \( m \) is the slope of the line and \( b \) is the y-intercept, the point where the line crosses \( y \) axis. The slope represents the rate at which \( y \) changes with respect to changes in \( x \) and can be positive or negative depending on the direction of the line.

For the equation \( y = 4(\frac{1}{2} x - 1) \), we can see it's a linear function because it can be simplified to \( y = 2x - 4 \), which fits into \( y = mx + b \) form with a slope \( m = 2 \) and y-intercept \( b = -4 \). Such functions are predictable and continuous, meaning the graph will be a straight line that extends infinitely in both directions on the plane.