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Sagot :
Sure! Let's address each problem step-by-step.
### Problem 24: Slope and Y-Intercept of the Line Containing Points (-1, 7) and (1, -3)
1. Calculate the slope (m):
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((-1, 7)\)[/tex] and [tex]\((1, -3)\)[/tex]:
[tex]\[ m = \frac{-3 - 7}{1 - (-1)} = \frac{-10}{2} = -5 \][/tex]
So, the slope of the line is:
[tex]\[ m = -5 \][/tex]
2. Find the y-intercept (b):
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
To find the y-intercept [tex]\( b \)[/tex], we can use one of the points. Let's use [tex]\((-1, 7)\)[/tex]:
[tex]\[ 7 = (-5)(-1) + b \][/tex]
Simplifying this:
[tex]\[ 7 = 5 + b \quad \Rightarrow \quad b = 7 - 5 = 2 \][/tex]
So the y-intercept is:
[tex]\[ b = 2 \][/tex]
Summing up, the equation of the line passing through [tex]\((-1, 7)\)[/tex] and [tex]\((1, -3)\)[/tex] is:
[tex]\[ y = -5x + 2 \][/tex]
### Problem 25: Point of Intersection of the Lines [tex]\(5x - 2y = -1\)[/tex] and [tex]\(4x - y = 1\)[/tex]
We need to find the point [tex]\((x, y)\)[/tex] where the lines intersect. This can be done by solving the system of linear equations.
1. Write the two equations:
[tex]\[ \begin{aligned} 5x - 2y &= -1 \quad \text{(1)} \\ 4x - y &= 1 \quad \text{(2)} \end{aligned} \][/tex]
2. Solve Equation (2) for [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 1 \][/tex]
3. Substitute [tex]\( y \)[/tex] into Equation (1):
[tex]\[ 5x - 2(4x - 1) = -1 \][/tex]
Simplify this equation:
[tex]\[ 5x - 8x + 2 = -1 \\ -3x + 2 = -1 \\ -3x = -3 \\ x = 1 \][/tex]
4. Find [tex]\( y \)[/tex] using [tex]\( x = 1 \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] back into [tex]\( y = 4x - 1 \)[/tex]:
[tex]\[ y = 4(1) - 1 = 4 - 1 = 3 \][/tex]
So, the point of intersection is:
[tex]\[ (x, y) = (1, 3) \][/tex]
### Problem 26: A rhombus is shown at the
It seems the problem statement for problem 26 is incomplete and does not provide enough context to proceed. To address this question thoroughly, more information is needed. If you have more details or a specific question about the rhombus, please provide those so that I can assist you better!
If you have any other math questions or specific queries, feel free to ask!
### Problem 24: Slope and Y-Intercept of the Line Containing Points (-1, 7) and (1, -3)
1. Calculate the slope (m):
The formula for the slope [tex]\( m \)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((-1, 7)\)[/tex] and [tex]\((1, -3)\)[/tex]:
[tex]\[ m = \frac{-3 - 7}{1 - (-1)} = \frac{-10}{2} = -5 \][/tex]
So, the slope of the line is:
[tex]\[ m = -5 \][/tex]
2. Find the y-intercept (b):
The equation of a line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
To find the y-intercept [tex]\( b \)[/tex], we can use one of the points. Let's use [tex]\((-1, 7)\)[/tex]:
[tex]\[ 7 = (-5)(-1) + b \][/tex]
Simplifying this:
[tex]\[ 7 = 5 + b \quad \Rightarrow \quad b = 7 - 5 = 2 \][/tex]
So the y-intercept is:
[tex]\[ b = 2 \][/tex]
Summing up, the equation of the line passing through [tex]\((-1, 7)\)[/tex] and [tex]\((1, -3)\)[/tex] is:
[tex]\[ y = -5x + 2 \][/tex]
### Problem 25: Point of Intersection of the Lines [tex]\(5x - 2y = -1\)[/tex] and [tex]\(4x - y = 1\)[/tex]
We need to find the point [tex]\((x, y)\)[/tex] where the lines intersect. This can be done by solving the system of linear equations.
1. Write the two equations:
[tex]\[ \begin{aligned} 5x - 2y &= -1 \quad \text{(1)} \\ 4x - y &= 1 \quad \text{(2)} \end{aligned} \][/tex]
2. Solve Equation (2) for [tex]\( y \)[/tex]:
[tex]\[ y = 4x - 1 \][/tex]
3. Substitute [tex]\( y \)[/tex] into Equation (1):
[tex]\[ 5x - 2(4x - 1) = -1 \][/tex]
Simplify this equation:
[tex]\[ 5x - 8x + 2 = -1 \\ -3x + 2 = -1 \\ -3x = -3 \\ x = 1 \][/tex]
4. Find [tex]\( y \)[/tex] using [tex]\( x = 1 \)[/tex]:
Substitute [tex]\( x = 1 \)[/tex] back into [tex]\( y = 4x - 1 \)[/tex]:
[tex]\[ y = 4(1) - 1 = 4 - 1 = 3 \][/tex]
So, the point of intersection is:
[tex]\[ (x, y) = (1, 3) \][/tex]
### Problem 26: A rhombus is shown at the
It seems the problem statement for problem 26 is incomplete and does not provide enough context to proceed. To address this question thoroughly, more information is needed. If you have more details or a specific question about the rhombus, please provide those so that I can assist you better!
If you have any other math questions or specific queries, feel free to ask!
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