Question

x and y intercepts worksheet (day 2)

1. if the x - intercept of a line is positive and the y - intercept is negative, does the line slant upward or downward from left to right? explain your reasoning.
2. a student says that the x - intercept of the graph x + 2y = 5 is the point (0,5). why is the student incorrect?
3. at which point does the graph of the equation 2x + y = 4 cross the x - axis?
4. what is the y - intercept of the graph of the equation 3x + y = 6?

Explanation:

1.

Step1: Recall the formula for slope using intercepts

The slope \(m\) of a line with \(x -\)intercept \(a\) (\(a>0\)) and \(y -\)intercept \(b\) (\(b < 0\)) can be found using the two - point formula. The two points on the line are \((a,0)\) and \((0,b)\). The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).

Step2: Calculate the slope

Substitute \(x_1=a,y_1 = 0,x_2=0,y_2=b\) into the slope formula. Then \(m=\frac{b - 0}{0 - a}=\frac{b}{-a}\). Since \(a>0\) and \(b < 0\), \(\frac{b}{-a}<0\).

Step1: Recall the definition of \(x -\)intercept

The \(x -\)intercept of a line is the point \((x,0)\) where the line crosses the \(x -\)axis.

Step2: Find the \(x -\)intercept of \(x + 2y=5\)

To find the \(x -\)intercept, set \(y = 0\). Then \(x+2(0)=5\), which gives \(x = 5\). The \(x -\)intercept is the point \((5,0)\), not \((0,5)\) (which is the \(y -\)intercept when \(x = 0\): \(0+2y=5\Rightarrow y=\frac{5}{2}\)).

Step1: Recall the property of \(x -\)axis crossing

A line crosses the \(x -\)axis at the point \((x,0)\).

Step2: Substitute \(y = 0\) into the equation \(2x+y=4\)

Set \(y = 0\) in the equation \(2x + y=4\). We get \(2x+0=4\).

Step3: Solve for \(x\)

Divide both sides of the equation \(2x=4\) by \(2\). So \(x = 2\).

Answer:

The line slants downward from left - to - right. A negative slope (\(m<0\)) means that as \(x\) increases, \(y\) decreases.

2.