The table gives the coordinates of two points on a line in the xy-plane. The y-intercept of the line is (k-5,b) , where k and b are constants. What is the value of b?

1. Find the slope (\(m\)) of the line using the given points \(( k, 13) \) and \(( k+ 7, - 15) \):
   \[m = \frac { - 15 - 13} { ( k+ 7) - k} = \frac { - 28} { 7} = - 4\]
2. Use the slope-intercept form \(y = mx + c\). Substitute \(m = - 4\) and one of the points, say \(( k, 13) \), to find \(c\):
   \[13 = - 4k + c \implies c = 13 + 4k\]
3. The y-intercept is given as \(( k- 5, b) \). Substitute \(x = k- 5\) into the equation \(y = - 4x + c\):
   \[b = - 4( k- 5) + c\]
4. Substitute \(c = 13 + 4k\) into the equation:
   \[b = - 4( k- 5) + ( 13 + 4k) \]
   Simplify:
   \[b = - 4k + 20 + 13 + 4k = 33\]
   Thus, the value of \(b\) is 33.

Supplemental Knowledge

In algebra, understanding the concept of the y-intercept is crucial when dealing with linear equations. The y-intercept is the point where a line crosses the y-axis, which means the x-coordinate at this point is zero. The general form of a linear equation is \(y = mx + c\), where \(m\) represents the slope and \(c\) represents the y-intercept.
To find the y-intercept from two points on a line, you first need to determine the slope (m) using the formula:
\[m = \frac { y_ 2 - y_ 1} { x_ 2 - x_ 1} \]
Once you have the slope, you can use one of the points to solve for \(c\) by substituting \(x\), \(y\), and \(m\) into the linear equation and solving for \(c\).
For example, if you have points (1, 2) and (3, 4):

1. Calculate the slope: 
   \[m = \frac { 4 - 2} { 3 - 1} = 1\]
2. Use one point to find c:
   \[2 = 1( 1) + c \Rightarrow c = 1\]
   Thus, your equation becomes \(y = x + 1\), and your y-intercept (b) is 1.

Real-World Applications

Imagine you're designing a ramp for wheelchair access in a building, meeting specific guidelines regarding its steepness is key to ensure its safety and accessibility. By plotting two points representing different heights at set distances along your proposed ramp path, it will enable you to determine its slope and conform with regulations; its "y-intercept" would correspond with where your ramp begins relative to ground level.
Similar to financial planning, understanding trends that develop over time can be illustrated using linear equations. If you are tracking savings growth or investment returns with consistent contributions or rates of return over time, plotting these on a graph helps predict future values based on current ones.

To deepen your understanding of linear equations and intercepts further, explore UpStudy's live tutor question bank or AI-powered problem-solving services. These resources provide personalized assistance tailored to your learning needs.
For those tackling math problems like this one involving coordinates and intercepts, UpStudy’s Algebra calculator can be an invaluable tool. It simplifies complex calculations so you can focus on mastering concepts rather than getting bogged down by arithmetic details.
Discover how UpStudy can transform your learning experience today!