Question

determine the equation of each line from the given representation.

a.

graph of a line on a coordinate plane with x-axis from -3 to 3 and y-axis from -3 to 3, intersecting y-axis at 3 and x-axis at 2 (approx)

b. a line with a slope $-\frac{2}{3}$ and passes through the point $(-3, 4)$.

c.

$x$	$-4$	$-3$	$-2$	$-1$
$y$	$-11$	$-9$	$-7$	$-5$

Explanation:

Part a

Step1: Identify slope and y-intercept

From the graph, the line crosses the y-axis at \( (0, 3) \), so \( b = 3 \). To find the slope, use two points: \( (0, 3) \) and \( (2, 0) \). Slope \( m=\frac{0 - 3}{2 - 0}=-\frac{3}{2} \).

Step2: Write slope - intercept form

The slope - intercept form is \( y=mx + b \). Substituting \( m =-\frac{3}{2} \) and \( b = 3 \), we get \( y=-\frac{3}{2}x + 3 \).

Step1: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \( (x_1,y_1)=(-3,4) \) and \( m =-\frac{2}{3} \).

Step2: Substitute values and simplify

Substitute into the formula: \( y - 4=-\frac{2}{3}(x + 3) \). Expand: \( y - 4=-\frac{2}{3}x-2 \). Add 4 to both sides: \( y=-\frac{2}{3}x + 2 \).

Step1: Calculate the slope

Use two points, e.g., \( (-4,-11) \) and \( (-3,-9) \). Slope \( m=\frac{-9-(-11)}{-3-(-4)}=\frac{2}{1}=2 \).

Step2: Find the y - intercept

Use the slope - intercept form \( y=mx + b \). Substitute \( x=-4 \), \( y = - 11 \), and \( m = 2 \): \( -11=2(-4)+b \). Solve for \( b \): \( -11=-8 + b\Rightarrow b=-3 \).

Step3: Write the equation

Substitute \( m = 2 \) and \( b=-3 \) into \( y=mx + b \), we get \( y = 2x-3 \).

Answer:

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

Part b