Problem 22 - Algebra 1

Question

Find three different ordered pairs that are solutions of the equation. \(y=7-4 x\)

Step-by-Step Solution

Answer

The three ordered pairs that are solutions for the equation \(y=7-4x\) are (0,7), (1,3), and (2,-1).

Step 1: Choose three distinct values for \(x\)

Let's choose \(x=0\), \(x=1\), and \(x=2\) as three distinct values for the variable \(x\).

Step 2: Substitute \(x\) into the equation and solve for \(y\)

For \(x=0\), the equation becomes \(y=7-4(0)\) which simplifies to \(y=7\). Therefore, the ordered pair for \(x=0\) is (0,7). For \(x=1\), the equation becomes \(y=7-4(1)\), simplifying to \(y=3\). Therefore, the ordered pair for \(x=1\) is (1,3). For \(x=2\), the equation becomes \(y=7-4(2)\), simplifying to \(y=-1\). Therefore, the ordered pair for \(x=2\) is (2,-1).

Key Concepts

Ordered Pairs

Ordered pairs are a fundamental concept in mathematics, especially in the context of coordinate geometry and graphing equations. An ordered pair consists of two elements, usually written in parentheses like this:

• (x, y)

The first element, x, represents a position along the horizontal axis, also known as the x-axis. The second element, y, indicates a position along the vertical axis, or the y-axis.
When we talk about solutions to equations, an ordered pair (x, y) is a solution if it, when substituted back into the equation, makes the equation true. For example, in the equation:

• \( y = 7 - 4x \)

An ordered pair like (0, 7) is a solution, because if we substitute x with 0, it results in y being 7, thereby balancing the equation. Finding ordered pairs is like finding points on a graph that satisfy the equation. This is key for visualizing and understanding relationships in linear equations.

Substitution Method

The substitution method is a powerful tool used to solve equations and systems of equations. The idea is to solve one of the equations for one variable and then substitute this expression into another equation. This method simplifies the problem by reducing the number of variables you need to deal with at one time.
In the context of finding ordered pairs for the equation:

• \( y = 7 - 4x \)

You would choose a value for x (a simple number like 0, 1, or 2), and substitute it back into the equation to solve for y. For example, substituting x = 1 gives:

• \( y = 7 - 4(1) = 3 \)

This simple step-by-step approach is helpful in understanding how variables relate each other in an equation. In more complex systems, this technique can be used alongside others to find more elaborate solutions.

Solving Equations

Solving equations is a core skill in mathematics that involves finding the values of variables that make an equation true. The equation identifies a relationship between the variables. The goal is to isolate the variable of interest on one side of the equation.
Take the equation:

• \( y = 7 - 4x \)

This is a linear equation, which means it will graph as a straight line. Solving for different values of x helps us find corresponding y values, providing enough points to sketch the line on a graph. This process not only finds solutions but also helps in better understanding the relationship represented by the equation.
By practicing solving such equations, confidence grows in handling both simple and complex mathematical problems. Many real-world scenarios require such problem-solving techniques, making this an essential ability in numerous fields and everyday situations.