1. How many solutions does the linear system have ?
3x - y = 4
-3x + 11y = 2
2. A line passes through the points (6,4) and (5,-3). What is the equation of the line in point-slope form ?
a. x - 6 = 7(y-4)
b. y - 4 = 7(x-6)
c. y + 6 = 7(x+4)
d. y - 6 = 7(x-4)
3. A line passes through the point (4,-3) and has a slope of -3. What is the equation of the line in standard form ?
4. Which point is a solution of the system of linear inequalities ?
y < -2x
y > 3x+5
a. (2,-1) b.(-4,1) c. (-1,4) d.(1,-2)
5. To solve the linear system below, which substitution of unkowns is proper?
a.sub 5x - 16 for y in the first Eqn
b.sub 5x + 12 for y in the first Eqn
c.sub 5x + 16 for y in the first Eqn
d.sub 7y - 4 for x in the second Eqn
PLEASE HELP ME.
1. since the lines are not parallel, they must intersect somewhere. And only one somewhere.
2.
(y-4)/(x-6) = 7/1
y-4 = 7(x-6)
3.
(y+3) = -3(x-4)
3x-y = -9
4.
y < -2x
y > 3x+5
-2x > y > 3x+5
-2x > 3x+5
-1 > x
x < -1, so (-4,1)
5. ?
Of course! I'll be happy to help you with your questions. Let's go through each question step by step and explain how to find the answers.
1. How many solutions does the linear system have?
To determine the number of solutions for the linear system, we can use the concept of determinants. The linear system can have three possibilities: one unique solution, no solution, or infinitely many solutions. To find the determinant, we need to set up the system of equations in matrix form:
| 3 -1 | | x | | 4 |
|-3 11 | x | y | = | 2 |
Calculate the determinant by multiplying the diagonals and subtracting the product of the off-diagonals:
Determinant = (3 * 11) - (-1 * -3) = 33 - 3 = 30
Now, if the determinant is non-zero (not equal to 0), then the system has a unique solution. In this case, since the determinant is non-zero (30 ≠ 0), the linear system has one unique solution.
2. A line passes through the points (6,4) and (5,-3). What is the equation of the line in point-slope form?
To determine the equation of a line in point-slope form, we need one point on the line and its slope. We have the points (6,4) and (5,-3), so let's calculate the slope first:
Slope (m) = (y2 - y1) / (x2 - x1)
= (-3 - 4) / (5 - 6)
= -7 / -1
= 7
We now have the slope. To find the equation of the line using point-slope form, we can use the equation:
y - y1 = m(x - x1)
Substituting the known values (6,4) and the slope (m = 7) into the equation:
y - 4 = 7(x - 6)
Therefore, the equation of the line in point-slope form is: y - 4 = 7(x - 6).
So, the answer is b: y - 4 = 7(x-6).
3. A line passes through the point (4,-3) and has a slope of -3. What is the equation of the line in standard form?
To find the equation of a line in standard form, we need to use the given point (4,-3) and slope (-3). The standard form of a linear equation is Ax + By = C, where A, B, and C are integers.
We can start by substituting the slope (-3) into the point-slope form equation:
y - y1 = m(x - x1)
y - (-3) = -3(x - 4)
y + 3 = -3x + 12
3x + y = 12 - 3
3x + y = 9
Therefore, the equation of the line in standard form is 3x + y = 9.
4. Which point is a solution of the system of linear inequalities?
The system of linear inequalities is y < -2x and y > 3x + 5. To determine which point satisfies both inequalities, we can substitute the x and y values of each point into both inequalities and check if they hold true.
Let's check the points one by one:
a. (2,-1)
For y < -2x: -1 < -2(2) => -1 < -4 (False)
For y > 3x + 5: -1 > 3(2) + 5 => -1 > 11 (False)
b. (-4,1)
For y < -2x: 1 < -2(-4) => 1 < 8 (True)
For y > 3x + 5: 1 > 3(-4) + 5 => 1 > -7 (True)
c. (-1,4)
For y < -2x: 4 < -2(-1) => 4 < 2 (False)
For y > 3x + 5: 4 > 3(-1) + 5 => 4 > 2 (True)
d. (1,-2)
For y < -2x: -2 < -2(1) => -2 < -2 (False)
For y > 3x + 5: -2 > 3(1) + 5 => -2 > 8 (False)
Based on these calculations, the point that satisfies both inequalities is b. (-4,1).
5. To solve the linear system below, which substitution of unknowns is proper?
a.sub 5x - 16 for y in the first equation
b.sub 5x + 12 for y in the first equation
c.sub 5x + 16 for y in the first equation
d.sub 7y - 4 for x in the second equation
Unfortunately, you did not provide the linear system. Please provide the equations, and I will guide you through the appropriate substitution method.
I hope this helps! Let me know if you have any further questions.