Question: Find the $ x $-intercept of the line that passes through the points $ (2, 5) $ and $ (6, 13) $. - Deep Underground Poetry
Find the $ x $-intercept of the line that passes through the points $ (2, 5) $ and $ (6, 13) $
Find the $ x $-intercept of the line that passes through the points $ (2, 5) $ and $ (6, 13) $
Understanding where a line meets the $ x $-axis—commonly called the $ x $-intercept—is a fundamental concept in math, especially in algebra and data trends. Even if the topic feels simple, grasping it deepens your analytical skills in interpreting real-world data. That’s why today, we explore how to find the $ x $-intercept of the line connecting two key points: $ (2, 5) $ and $ (6, 13) $. Whether you’re reviewing equations, interpreting graphs, or analyzing growth patterns, this foundational skill helps clarify how variables relate in practical scenarios.
Why the $ x $-Intercept Matters Across Domains
Understanding the Context
In educational and professional environments, the $ x $-intercept—where $ y = 0 $—reveals critical insights. In economics, it may represent the break-even point where profit vanishes. In environmental science, it can indicate when a pollutant concentration drops to zero. For students learning linear equations, visualizing where lines cross axes builds spatial reasoning and strengthens problem-solving confidence. Beyond textbooks, recognition of this concept fuels curiosity around data trends and predictive modeling—tools increasingly central to digital literacy in the US market.
How to Compute the $ x $-Intercept Step-by-Step
To find the $ x $-intercept, locate the point on the line where $ y = 0 $. This requires first determining the line’s equation using two known points: $ (2, 5) $ and $ (6, 13) $. Start by calculating the slope, or rate of change, given by $ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2 $. With slope $ m = 2 $ and a point like $ (2, 5) $, apply the point-slope form: $ y - y_1 = m(x - x_1) $. Substituting values: $ y - 5 = 2(x - 2) $. Expanding gives $ y = 2x - 4 + 5 $, so $ y = 2x + 1 $.
Now set $ y = 0 $ to solve for the $ x $-intercept:
$ 0 = 2x + 1 $ → $ 2x = -1 $ → $ x = -\frac{1}{2} $. Thus, the line crosses the $ x $-axis at $ \left( -\frac{1}{2}, 0 \right) $. This method—calculating slope, writing the equation, and solving for $ y
