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When we talk about graphing solutions for a linear equation like \( x + 5y = 5 \), our goal is to showcase all the points (also known as ordered pairs) that satisfy the equation on a graph. Here's how it works: start by selecting various values for one of the variables, usually \( y \), and calculate the corresponding \( x \) values using the equation. These values (\( x, y \)) form the points that you will plot on a graph.

Once you have these points plotted, take a straight edge, like a ruler, to draw a line through these points. This line is not just any line but a graphical representation of the equation \( x + 5y = 5 \).

• The line continues infinitely in both directions in a straight path.
• Every point along the line is a solution to the equation.
• Any point not on the line does not satisfy the equation.

Graphing solutions help in visualizing the relationship between two variables and understanding how changes in one affect the other.

An ordered pair is a set of two numbers that describe the position of a point on a graph. In algebra, an ordered pair often comes in the form of \((x, y)\). For our equation \( x + 5y = 5 \), finding ordered pairs involves selecting a value for \( y \) and computing the corresponding \( x \) value using the rearranged equation, \( x = 5 - 5y \).

Each chosen value of \( y \) gives a unique \( x \), resulting in different points on the graph.

• If \( y = 0 \), \( x = 5 \), thus the ordered pair is \((5, 0)\).
• If \( y = 1 \), \( x = 0 \), resulting in the ordered pair \((0, 1)\).
• And so forth for other values like \( (10, -1) \) and \( (15, -2) \).

These ordered pairs are critical because they provide specific points that, when plotted, reveal the linear relationship defined by the equation. They act as a blueprint for sketching the entire line on the graph.

In linear equations, understanding variables is crucial. Variables are symbols (often \( x \) and \( y \)) used to represent numbers in equations. For the equation \( x + 5y = 5 \), \( x \) and \( y \) are the variables. Their values fluctuate, making them dynamic elements in algebraic expressions.

In our specific equation, \( x \) depends on \( y \). When you choose different values for \( y \) (like we did in our ordered pairs), \( x \) changes accordingly. This dependency highlights the relational aspect of equations where:

• Changing \( y \) affects \( x \) based on the defined rule \( x = 5 - 5y \).
• Both variables are essential for illustrating the equation's solutions and corresponding graph.

In essence, variables are what allow equations to express relationships between different quantities, making them fundamental in math for modeling real-world situations.