- The slope of the line is \(m = 0.25\) (positive) and the \(y\)-intercept is \(b = -30\) (negative).
- For the given statements: (a) False, (b) True, (c) True, (d) True.
- The equation of the line is \(y = 0.25x - 30\).
- Final answers: a. False, b. True, c. True, d. True, Equation: \(y = 0.25x - 30\).
Explanation
- Analyzing Line j
The line \(j\) contains the points \((-40, -40)\), \((50, -17.5)\), \((80, -10)\), and \((110, -2.5)\). The slope \(m\) of a line passing through points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Using the points \((-40, -40)\) and \((50, -17.5)\):
\[m = \frac{-17.5 - (-40)}{50 - (-40)} = \frac{22.5}{90} = 0.25 = \frac{1}{4}.\]
Since the slope is constant, the line is indeed straight. To find the \(y\)-intercept \(b\), we use the equation \(y = mx + b\) with the point \((50, -17.5)\):
\[-17.5 = 0.25(50) + b\]
\[-17.5 = 12.5 + b\]
\[b = -17.5 - 12.5 = -30.\]
Thus, the equation of the line \(j\) is \(y = 0.25x - 30\).
- Evaluating Statements
Now we evaluate the statements provided in problem 4:
AThe slope of line \(j\) is \(0.25\), which is positive. Therefore, the statement "The slope of line \(j\) is negative" is False.
BThe \(y\)-intercept \(b\) is \(-30\), which is less than 0. Therefore, the statement "The \(y\)-intercept is negative" is True.
CTo check if line \(j\) passes through \((-10, 15)\), we substitute \(x = -10\) into the equation \(y = 0.25x - 30\):
\[y = 0.25(-10) - 30 = -2.5 - 30 = -32.5.\]
Since \(-32.5
eq 15\), the line does not pass through \((-10, 15)\). Thus, the statement "Line \(j\) does not pass through \((-10, 15)\)" is True.
DA line in the Cartesian plane represents the set of all points \((x, y)\) that satisfy a linear equation. Therefore, the statement "Line \(j\) represents a set of all solutions to an equation in two variables" is True.
- Final Results
The equation for line \(j\) is \(y = 0.25x - 30\). To graph this line, we can use two points from the table, such as \((80, -10)\) and \((110, -2.5)\). By plotting these points on a coordinate plane and drawing a straight line through them, we represent the set of all solutions to the equation. Thus, the equation is \(y = 0.25x - 30\). The True/False results for problem 4 are:
Examples
In real life, linear equations are essential for modeling consistent relationships. For example, if you are renting a piece of equipment for a flat fee plus an hourly rate, you can represent your total cost \(y\) after \(x\) hours using the equation \(y = mx + b\), where \(m\) is the hourly rate and \(b\) is the initial fee. Similarly, by calculating the slope from a few data points, you can predict costs or travel times, much like identifying the properties of line \(j\) allows us to understand its behavior and predict its path across the coordinate plane.