For each equation, (a) give a table with at least three ordered pairs that are solutions, and (b) graph the equation. See Examples 3 and 4.
2x + 3y = 5

Verified step by step guidance

1. Rewrite the given linear equation \$2x + 3y = 5\( to express \)y\( in terms of \)x\(. This will help in finding ordered pairs. To do this, isolate \)y\(: \)3y = 5 - 2x$, so \(y = \frac{5 - 2x}{3}\).
2. Choose at least three different values for \(x\). For each chosen \(x\) value, substitute it into the expression for \(y\) to find the corresponding \(y\) value. This will give you ordered pairs \((x, y)\) that satisfy the equation.
3. For example, if you pick \(x = 0\), substitute into \(y = \frac{5 - 2(0)}{3} = \frac{5}{3}\). So one ordered pair is \((0, \frac{5}{3})\). Repeat this for two more values of \(x\) (like \(x=3\) and \(x=-3\)) to get at least three ordered pairs.
4. Once you have the ordered pairs, create a table listing the \(x\) values and their corresponding \(y\) values. This table clearly shows the solutions to the equation.
5. To graph the equation, plot the ordered pairs on the coordinate plane. Then, draw a straight line through these points because the equation represents a linear function. This line is the graph of the equation \$2x + 3y = 5$.

Key Concepts

Linear Equations in Two Variables

A linear equation in two variables, like 2x + 3y = 5, represents a straight line on the coordinate plane. Each solution is an ordered pair (x, y) that satisfies the equation. Understanding how to find these pairs is essential for graphing and analyzing the line.

Finding Ordered Pairs (Solutions)

To find ordered pairs that satisfy the equation, assign values to one variable and solve for the other. For example, choosing x-values and calculating corresponding y-values creates points that lie on the line, which can be tabulated for clarity.

Graphing Linear Equations

Graphing involves plotting the ordered pairs on the coordinate plane and connecting them to form a straight line. This visual representation helps understand the relationship between variables and the set of all possible solutions.