Question

write an equation of the line in slope-i

1. \\( m = -\frac{2}{5} \\)

(grid with x from -6 to 6, y from -6 to 6, a point at (-5, -2), and a line segment. then blanks: \\( = -\frac{2}{5} \\), \\( b = \\) ____, equation: ____)

Explanation:

Step1: 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 $m =-\frac{2}{5}$. We need to find the value of $b$.

We can use a point on the line to find $b$. From the graph, we can see that the line passes through the point $(x,y)=(1,2)$ (we can also use the point $(- 5,-2)$). Let's use the point $(1,2)$ and the slope $m =-\frac{2}{5}$ in the equation $y=mx + b$.

Substitute $x = 1$, $y=2$ and $m=-\frac{2}{5}$ into $y=mx + b$:

$2=-\frac{2}{5}(1)+b$

Step2: Solve for $b$

First, simplify the right - hand side: $2=-\frac{2}{5}+b$

To solve for $b$, add $\frac{2}{5}$ to both sides of the equation:

$b=2+\frac{2}{5}=\frac{10 + 2}{5}=\frac{12}{5}=2.4$? Wait, no, let's check the point $(-5,-2)$. Let's substitute $x=-5$, $y = - 2$ and $m=-\frac{2}{5}$ into $y=mx + b$:

$-2=-\frac{2}{5}(-5)+b$

Simplify the right - hand side: $-\frac{2}{5}\times(-5)=2$, so the equation becomes $-2 = 2 + b$

Subtract 2 from both sides: $b=-2 - 2=-4$? Wait, that's wrong. Wait, let's look at the graph again. The line passes through $(1,2)$? Wait, no, the dot is at $(-5,-2)$. Let's use the slope formula. The slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let's take two points. Let's assume the y - intercept is at $(0,b)$. Let's take the point $(-5,-2)$ and $(0,b)$. The slope $m=\frac{b-(-2)}{0 - (-5)}=\frac{b + 2}{5}$. We know that $m=-\frac{2}{5}$, so:

$\frac{b + 2}{5}=-\frac{2}{5}$

Multiply both sides by 5: $b + 2=-2$

Subtract 2 from both sides: $b=-4$? Wait, no, when $x = 0$, what is $y$? Wait, let's check the line. If $m=-\frac{2}{5}$, starting from the point $(-5,-2)$, if we move 5 units to the right (since the denominator of the slope is 5) and 2 units up (since the numerator is - 2, wait, slope is rise over run, $m=\frac{\text{rise}}{\text{run}}=-\frac{2}{5}$, so run is 5 (positive x - direction), rise is - 2 (negative y - direction) or run is - 5 (negative x - direction), rise is 2 (positive y - direction).

From the point $(-5,-2)$, if we move 5 units to the right (x becomes $-5 + 5=0$) and 2 units up (y becomes $-2+2 = 0$)? No, that's not right. Wait, maybe the point on the line is $(1,2)$? Let's check the slope between $(-5,-2)$ and $(1,2)$: $m=\frac{2-(-2)}{1-(-5)}=\frac{4}{6}=\frac{2}{3}$, which is not $-\frac{2}{5}$. So I made a mistake. Wait, the slope is given as $m =-\frac{2}{5}$. Let's look at the line in the graph. The line passes through $(0,0)$? No, the line has a slope of $-\frac{2}{5}$. Wait, maybe the y - intercept is 0? No, let's re - examine.

Wait, the problem says "Write an equation of the line in slope - intercept form". The slope $m=-\frac{2}{5}$. Let's find two points on the line. Let's see, when $x = 0$, what is $y$? Let's use the slope formula. Let's take a point $(x_1,y_1)$ and $(x_2,y_2)$ on the line. Suppose we take $x = 0$, $y=b$ (the y - intercept) and another point. Let's assume the line passes through $(5,-2)$? Wait, $m=\frac{-2 - b}{5-0}=-\frac{2}{5}$. Then $\frac{-2 - b}{5}=-\frac{2}{5}$, multiply both sides by 5: $-2 - b=-2$, so $-b = 0$, $b = 0$? No, that's not matching the graph. Wait, maybe the point is $(1,2)$ is not on the line. Wait, the graph has a line with slope $-\frac{2}{5}$. Let's start from the origin $(0,0)$. If $m=-\frac{2}{5}$, then for $x = 5$, $y=-\frac{2}{5}\times5=-2$, so the point $(5,-2)$ is on the line. And for $x=-5$, $y=-\frac{2}{5}\times(-5)=2$, so the point $(-5,2)$? But the dot is at $(-5,-2)$. Wait, maybe there is a mistake in my point selection.

Wait, the given slope is $m =-\frac{2}{5}$. Let's use the point - slope form $y - y_1=m(x - x_1)$ with the point $(-5,-2)$. So $y-(-2)=-\frac{2}{5}(x - (-5))$, which simplifies to $y + 2=-\frac{2}{5}(x + 5)$

Expand the right - hand side: $y+2=-\frac{2}{5}x-2$

Subtract 2 from both sides: $y=-\frac{2}{5}x-4$

Ah! So when $x = 0$, $y=-4$, so the y - intercept $b=-4$? Wait, no, $y=-\frac{2}{5}x-4$? Let's check with $x=-5$: $y=-\frac{2}{5}\times(-5)-4=2 - 4=-2$, which matches the point $(-5,-2)$. And with $x = 5$: $y=-\frac{2}{5}\times5-4=-2 - 4=-6$, does that match the graph? The graph shows that at $x = 5$, $y$ is around - 6? Maybe. So the slope $m=-\frac{2}{5}$, and $b=-4$? Wait, no, when $x = 0$, $y=b$. From the equation $y=-\frac{2}{5}x-4$, when $x = 0$, $y=-4$, so $b=-4$.

Wait, let's do it correctly. The slope - intercept form is $y=mx + b$. We know $m=-\frac{2}{5}$. Let's take the point $(-5,-2)$ and substitute into $y=mx + b$:

$-2=-\frac{2}{5}(-5)+b$

Simplify $-\frac{2}{5}(-5)=2$, so:

$-2 = 2 + b$

Subtract 2 from both sides: $b=-2 - 2=-4$

Yes, that's correct. So $m=-\frac{2}{5}$ and $b=-4$

Then the equation of the line is $y=-\frac{2}{5}x-4$

Answer:

$m =-\frac{2}{5}$, $b=-4$, Equation: $y =-\frac{2}{5}x-4$