Comparing the equations $$y = 4x$$ and $$y = 4$$ highlights the difference between an equation with two variables and an equation with a single variable. In the two-variable equation $$y = 4x$$, the value of $$y$$ depends directly on the chosen value of $$x$$, meaning as the $$x$$-coordinate changes, the $$y$$-coordinate changes accordingly; visually, this produces a slanted line graph. Conversely, the one-variable equation $$y = 4$$ specifies that $$y$$ is a constant value independent of $$x$$; because the $$y$$-coordinate is always $$4$$, graphing this equation yields a straight horizontal line.

Claude

A **horizontal line** represents the graph of a linear equation that takes the form $$y = b$$, containing only the variable $$y$$ and no $$x$$ (e.g., $$y = 4$$). In this scenario, the $$y$$-value is a strict constant and does not depend on the shifting value of $$x$$. Regardless of the chosen $$x$$-value, the $$y$$-coordinate is invariably equal to $$b$$. Therefore, this line runs flat across the coordinate plane and passes through the $$y$$-axis precisely at the point $$(0, b)$$. To translate this into a graph via the point-plotting method, one constructs a table of values by maintaining the constant $$b$$ for all $$y$$-coordinates and selecting any variety of numbers for the $$x$$-coordinates.

Horizontal Line

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To graph the horizontal line equation $$y = 4$$, note that the equation lacks the variable $$x$$ and asserts that the $$y$$-value is a constant $$4$$. Because $$y$$ does not depend on $$x$$, any value can be chosen for the $$x$$-coordinates when constructing a table of values using the Point-Plotting Method. Fill in $$4$$ for all the $$y$$-coordinates and select convenient numbers, such as $$0$$, $$2$$, and $$4$$, for the $$x$$-coordinates. This creates the ordered pairs $$(0, 4)$$, $$(2, 4)$$, and $$(4, 4)$$. Plotting these points on a coordinate plane and connecting them yields a horizontal line that passes through the $$y$$-axis at $$(0, 4)$$.

Graphing $$y = 4$$ by Plotting Points

To graph the equation $$y = -1$$, recognize that it features only one variable, indicating the $$y$$-value is a constant $$-1$$ no matter what the $$x$$-value is. To proceed, build a table of values where every $$y$$-coordinate is $$-1$$, and pick arbitrary values for the $$x$$-coordinates (such as $$0$$, $$3$$, and $$-3$$), producing ordered pairs like $$(0, -1)$$, $$(3, -1)$$, and $$(-3, -1)$$. Plot these ordered pairs on the coordinate plane and draw a line connecting them. Consistent with equations of this form, the resulting graph is a horizontal line intersecting the $$y$$-axis at $$-1$$.

Graphing $$y = -1$$ by Plotting Points

Comparison of $$y = 4x$$ and $$y = 4$$

Graphing the equations $$y = -3x$$ and $$y = -3$$ on the same rectangular coordinate system clearly demonstrates the differences between linear equations with two variables versus a single variable. For the two-variable equation $$y = -3x$$, the value of $$y$$ depends entirely on the chosen value of $$x$$. Generating a table of values by selecting arbitrary $$x$$-coordinates yields scattered ordered pairs such as $$(0, 0)$$, $$(1, -3)$$, and $$(2, -6)$$. Plotting these points produces a slanted line that passes through the origin. Conversely, the single-variable equation $$y = -3$$ lacks an $$x$$ variable, meaning the $$y$$-value remains a constant $$-3$$ regardless of any chosen $$x$$-coordinate. Creating a table of values for this equation results in ordered pairs like $$(0, -3)$$, $$(1, -3)$$, and $$(2, -3)$$. Plotting these points produces a straight horizontal line that intersects the $$y$$-axis at $$-3$$.

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