Problem 37 - Algebra 1: Concepts and Skills

Question

Find three ordered pairs that are solutions of the equation. $$ 5 x+2 y=10 $$

Step-by-Step Solution

Answer

The three solutions are (0, 5), (2, 0), and (1, 2.5).

Step 1: Pick a Value for x

Let's start with a simple example by choosing \(x = 0\). This gives us the simplified equation \(2y = 10\).

Step 2: Solve for y

Dividing both sides of the equation by 2, we find that \(y = 5\). So one solution to the equation is the ordered pair \((0, 5)\).

Step 3: Find the Second Pair

Similarly, we can find a second solution by setting \(x = 2\). Then we have \(5*2 + 2y = 10\), or equivalently, \(2y = 10 - 10\). Simplifying, we see that \(2y = 0\) which means that \(y = 0\). So a second solution is the ordered pair \((2, 0)\).

Step 4: Find the Third Pair

Now, let's try \(x = 1\). We get \(5*1 + 2y = 10\), or \(2y = 5\). After dividing both sides by 2, we find that \(y = 2.5\). So a third solution is the ordered pair \((1, 2.5)\).

Key Concepts

Ordered Pairs

An ordered pair consists of two elements represented in a specific order, typically enclosed in parentheses. In the context of geometry and algebra, ordered pairs are often used to denote coordinates on a Cartesian plane. They take the form

• The first element of an ordered pair, usually represented by "x," indicates the horizontal position or the "x-coordinate."
• The second element, represented by "y," indicates the vertical position or the "y-coordinate."

For example, in the ordered pair (x, y), "x" denotes how far right or left the point is from the origin, and "y" denotes how far up or down it is. Ordered pairs are fundamental when plotting points on a graph as they precisely position points within the two-dimensional plane.

Solutions of Equations

A solution to an equation is a set of values for the variables that make the equation true. For linear equations in two variables, the solution is often expressed as ordered pairs. This allows us to visually interpret the relationship between variables on a graph.

Finding solutions involves selecting a particular value for one variable and calculating the corresponding value for the other variable. The number of solutions that can be generated for a linear equation is infinite, as you can select countless different starting values.

• The process often starts with choosing easy values for one variable, such as 0 or 1, to simplify calculations.
• This approach results in solutions that can be interpreted as points on a graph.

In the original example, the solution (x = 0, y = 5) indicates that the point (0, 5) satisfies the equation. This value can be plugged back into the equation to check its validity.

Graphing Linear Equations

Graphing linear equations provides a visual representation of the solutions and the relationship between variables. Each solution to the equation, expressed as an ordered pair, corresponds to a specific point on the Cartesian plane.

To graph a linear equation, you typically follow these steps:

• Identify several solutions to the equation, these are your ordered pairs.
• Plot these ordered pairs as points on the graph.
• Draw a line through these points. The line represents all possible solutions to the equation.

For the equation (5x + 2y = 10), solutions like (0, 5), (2, 0), and (1, 2.5) help illustrate the equation's line. The graph of a linear equation is always a straight line, reflecting the consistent relationship between the two variables.