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A line that passes through the point $(x, y)$, with a $y$-intercept of $b$ and a slope of $m$, can be represented by the equation

$y=m x+b$.



A line is drawn on the coordinate plane that passes through the point $(10,1)$ and has a slope of -0.5. The $y$-intercept of the line is ___.

Verified Answer :

  • Start with the slope-intercept form \(y = mx + b\).
  • Substitute the given slope \(m = -0.5\) and the point \((10, 1)\) to get \(1 = -0.5(10) + b\).
  • Solve the equation for \(b\), yielding \(b = 6\).
  • The final answer is \(\boxed{6}\).

Explanation

  1. Analyzing the Equation and Substituting Given Values

To find the \(y\)-intercept \(b\) of a line with a given slope \(m\) that passes through a specific point \((x, y)\), we start with the slope-intercept form of the equation of a line:
\[y = mx + b.\]
We are given the following information:

  1. The line passes through the point \((10, 1)\), so \(x = 10\) and \(y = 1\).
  2. The slope of the line is \(m = -0.5\).

Substituting these values into the slope-intercept equation, we have:
\[1 = (-0.5)(10) + b.\]

  1. Solving for the y-intercept

Now we solve the equation for the unknown constant \(b\):
\[1 = -5 + b.\]
To isolate \(b\), we add \(5\) to both sides of the equation:
\[1 + 5 = b,\]
which simplifies to:
\[b = 6.\]
Thus, the \(y\)-intercept of the line is \(6\).

Examples

In real-world applications, such as analyzing the trend of a project's budget, the \(y\)-intercept represents the starting point or the fixed cost, while the slope represents the rate of change per unit of time or activity. For instance, if a company's research budget starts at \(\$6\) million (\(b=6\)) and decreases at a rate of \(\$0.5\) million per year (\(m=-0.5\)), after 10 years (\(x=10\)), the budget would be \(y = -0.5(10) + 6 = \$1\) million. Identifying the \(y\)-intercept is a fundamental step in predicting future values in linear models.

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