Question

algebra 1 mp #2 - part 2
points j and k lie on the same line in a coordinate plane, as the coordinate plane below (with points j(3, 3.5) and k(6, 5) plotted).
a. what is the slope of the line passing through j and k? show or explain all your work. (3 points)
b. write the equation of the line passing through j and k. show or explain all your work. (3 points)
c. solve the inequality ( 3 leq 2x + 1 ) (4 points)

Explanation:

Part A: Slope of the line through J and K

Step 1: Recall slope formula

The slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
Given \( J(3, 3.5) \) and \( K(6, 5) \), let \( (x_1, y_1) = (3, 3.5) \) and \( (x_2, y_2) = (6, 5) \).

Step 2: Substitute values into formula

\( m = \frac{5 - 3.5}{6 - 3} = \frac{1.5}{3} = 0.5 \) or \( \frac{1}{2} \).

Step 1: Use point - slope form

The point - slope form of a line is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. We know \( m=\frac{1}{2} \) and we can use the point \( J(3, 3.5) \).

Step 2: Substitute into point - slope form

\( y - 3.5=\frac{1}{2}(x - 3) \)

Step 3: Convert to slope - intercept form (\( y=mx + b \))

Expand the right - hand side: \( y-3.5=\frac{1}{2}x-\frac{3}{2} \) (since \( \frac{1}{2}\times3 = \frac{3}{2}=1.5 \))
Add \( 3.5 \) to both sides. \( 3.5=\frac{7}{2} \), so \( y=\frac{1}{2}x-\frac{3}{2}+\frac{7}{2} \)
Simplify the right - hand side: \( y=\frac{1}{2}x+\frac{- 3 + 7}{2}=\frac{1}{2}x+\frac{4}{2}=\frac{1}{2}x + 2 \)

We can also verify with point \( K(6,5) \). Substitute \( x = 6 \) into \( y=\frac{1}{2}x + 2 \): \( y=\frac{1}{2}\times6+2=3 + 2=5 \), which matches the \( y \) - coordinate of \( K \).

Step 1: Subtract 1 from both sides

Subtract 1 from each side of the inequality \( 3\leq2x + 1 \): \( 3-1\leq2x+1 - 1 \)
Simplify: \( 2\leq2x \)

Step 2: Divide both sides by 2

Divide each side by 2: \( \frac{2}{2}\leq\frac{2x}{2} \)
Simplify: \( 1\leq x \) or \( x\geq1 \)

Answer:

The slope of the line passing through \( J \) and \( K \) is \( \frac{1}{2} \) (or \( 0.5 \)).

Part B: Equation of the line through J and K