Certainly! Let's work through each question step by step to find the equations of the lines described.
### Question 22
We need to write the equation of the line that passes through the point [tex]\((2, -11)\)[/tex] and has a slope of [tex]\(-\frac{5}{8}\)[/tex].
To find the equation, we can use the point-slope form of a line's equation, which is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
where [tex]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\(m\)[/tex] is the slope.
Here, [tex]\((x_1, y_1) = (2, -11)\)[/tex] and [tex]\(m = -\frac{5}{8}\)[/tex]. Plugging these values into the point-slope formula, we get:
[tex]\[ y - (-11) = -\frac{5}{8}(x - 2) \][/tex]
Simplifying the left side:
[tex]\[ y + 11 = -\frac{5}{8}(x - 2) \][/tex]
So, the equation of the line is:
[tex]\[ y - (-11) = -0.625(x - (2)) \][/tex]
### Question 23
We need to write the equation of the line that has a slope of 4 and a [tex]\(y\)[/tex]-intercept of -3.
The slope-intercept form of a line's equation is:
[tex]\[ y = mx + b \][/tex]
where [tex]\(m\)[/tex] is the slope and [tex]\(b\)[/tex] is the [tex]\(y\)[/tex]-intercept.
Here, [tex]\(m = 4\)[/tex] and [tex]\(b = -3\)[/tex]. Plugging in these values, we get:
[tex]\[ y = 4x + (-3) \][/tex]
So, the equation of the line is:
[tex]\[ y = 4x - 3 \][/tex]
### Question 24
We need to write the equation of the line that passes through the points [tex]\((-1, 7)\)[/tex] and [tex]\((1, -3)\)[/tex].
First, we need to find the slope [tex]\(m\)[/tex] of the line using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
where [tex]\((x_1, y_1) = (-1, 7)\)[/tex] and [tex]\((x_2, y_2) = (1, -3)\)[/tex]. Plugging in these values, we get:
[tex]\[ m = \frac{-3 - 7}{1 - (-1)} = \frac{-10}{2} = -5 \][/tex]
Now, we can use the point-slope form of the line's equation with one of the points, say [tex]\((-1, 7)\)[/tex]:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Substituting [tex]\((x_1, y_1) = (-1, 7)\)[/tex] and [tex]\(m = -5\)[/tex], we get:
[tex]\[ y - 7 = -5(x - (-1)) \][/tex]
Simplifying the right side:
[tex]\[ y - 7 = -5(x + 1) \][/tex]
So, the equation of the line is:
[tex]\[ y - (7) = -5.0(x - (-1)) \][/tex]
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In summary, the equations of the lines for each question are:
22. [tex]\( y + 11 = -\frac{5}{8}(x - 2) \)[/tex] or [tex]\( y - (-11) = -0.625(x - (2)) \)[/tex]
23. [tex]\( y = 4x - 3 \)[/tex]
24. [tex]\( y - 7 = -5(x + 1) \)[/tex] or [tex]\( y - (7) = -5.0(x - (-1)) \)[/tex]
