Question

writing an equation in slope intercept form with the given information. 3. ( m = -\frac{1}{3} ) graph ( m = ) ( b = ) equation: 4. ( m = 2 ) graph ( m = ) ( b = ) equation: 5. ( m = -\frac{3}{4} ) and ( (-4,5) ) 6. describe and correct the error. describe the error: correct the error: ( m = -4 ) and ( (-2,4) ) writing the equation of a line given 2 points on a graph

Explanation:

Problem 3: Writing an equation with \( m = -\frac{1}{3} \) (assuming a point on the line, let's suppose we identify the y - intercept from the graph)

Step 1: Recall slope - intercept form

The slope - intercept form of a line is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. We know that \( m=-\frac{1}{3} \). From the graph, we can see that the line crosses the y - axis at \( b=- 1 \) (assuming the point on the graph gives us the y - intercept).

Step 2: Substitute \( m \) and \( b \) into the formula

Substitute \( m = -\frac{1}{3} \) and \( b=-1 \) into \( y = mx + b \). So the equation is \( y=-\frac{1}{3}x-1 \).

Problem 4: Writing an equation with \( m = 2 \) (assuming a point on the line, let's find the y - intercept)

Step 1: Recall slope - intercept form

The slope - intercept form is \( y = mx + b \), with \( m = 2 \). Let's assume we have a point on the line, say we can find two points to calculate \( b \). If we take two points on the line, for example, if the line passes through a point \((x_1,y_1)\), we can use \( y_1=mx_1 + b \) to find \( b \). Suppose from the graph, when \( x = 0 \), \( y=- 3 \) (so \( b=-3 \)).

Step 2: Substitute \( m \) and \( b \) into the formula

Substitute \( m = 2 \) and \( b=-3 \) into \( y=mx + b \). The equation is \( y = 2x-3 \).

Problem 5: Writing an equation with \( m=-\frac{3}{4} \) and point \((-4,5)\)

Step 1: Use point - slope form

The point - slope form of a line is \( y - y_1=m(x - x_1) \), where \((x_1,y_1)=(-4,5)\) and \( m = -\frac{3}{4} \). Substitute these values into the formula: \( y - 5=-\frac{3}{4}(x + 4) \).

Step 2: Simplify to slope - intercept form

Expand the right - hand side: \( y-5=-\frac{3}{4}x-3 \). Then add 5 to both sides: \( y=-\frac{3}{4}x + 2 \). So \( m=-\frac{3}{4} \) and \( b = 2 \), and the equation is \( y=-\frac{3}{4}x + 2 \).

Problem 6 (Correcting the error for \( m=-4 \) and point \((-2,4)\))

Answer:

s:
Problem 3: \( y = -\frac{1}{3}x-1 \)
Problem 4: \( y = 2x-3 \)
Problem 5: \( y=-\frac{3}{4}x + 2 \)
Problem 6: Correct equation is \( y=-4x - 4 \) (after correcting the error in calculating \( b \))