A system of equations is given below.
y = negative 2 x + one-fourth and y = negative 2 x minus one-fourth
Which of the following statements best describes the two lines?
They have the same slope but different y-intercepts, so they have no solution.
They have the same slope but different y-intercepts, so they have one solution.
They have different slopes but the same y-intercept, so they have no solution.
They have different slopes but the same y-intercept, so they have one solution.
They have the same slope but different y-intercepts, so they have one solution.
The equations in the system below are equivalent.
y = 2 (x + 4). Y = 2 x + 8
How many solutions does the system have?
The system has infinite solutions because the two equations represent the same line. Any value of x plugged into either equation will result in the same value of y. Therefore, any ordered pair (x, y) that satisfies one equation will also satisfy the other equation.
How many solutions does this system have?
6 x + 3 y = negative 12. y = negative 2 x + 4.
We can solve this system using substitution or elimination method.
Using substitution, we can substitute y = -2x + 4 into the first equation:
6x + 3(-2x + 4) = -12
Simplifying the left side:
6x - 6x + 12 = -12
12 = -12
This is a contradiction, which means there are no solutions to this system.
Using elimination, we can multiply the second equation by 3 to get 3y = -6x + 12, and then add it to the first equation:
6x + 3y = -12
+3y = -6x + 12
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6x + 6y = 0
Dividing both sides by 6, we get:
x + y = 0
This is the equation of a line with a slope of -1 and a y-intercept of 0. Any point on this line (x, -x) will satisfy both equations. Therefore, there are infinite solutions to this system.
However, since the two methods give conflicting results, we can conclude that the system is inconsistent (has no solution).
The measures of two supplementary angles total 180 degrees. The measure of angle y is 65 degrees less than the measure of angle x. What are the measures of the angles?
The measure of angle x is 122.5 degrees. The measure of angle y is 57.5 degrees.
The measure of angle x is 115 degrees. The measure of angle y is 65 degrees.
The measure of angle x is 57.5 degrees. The measure of angle y is 122.5 degrees.
The measure of angle x is 65 degrees. The measure of angle y is 115 degrees.
Let x be the measure of angle x in degrees. Then, the measure of angle y is x - 65 (65 degrees less than x).
Since the angles are supplementary, we have:
x + (x - 65) = 180
Combining like terms:
2x - 65 = 180
Adding 65 to both sides:
2x = 245
Dividing both sides by 2:
x = 122.5
So, the measure of angle x is 122.5 degrees. Then, the measure of angle y is:
y = x - 65 = 122.5 - 65 = 57.5
Therefore, the measures of the angles are x = 122.5 degrees and y = 57.5 degrees.
The correct answer is: The measure of angle x is 122.5 degrees. The measure of angle y is 57.5 degrees.
Which system of equations below has exactly one solution?
A(y = –8x – 6 and y = –8x + 6
B(y = –8x – 6 and One-halfy = –4x – 3
C(y = –8x – 6 and y = 8x – 6
D(y = –8x – 6 and –y = 8x + 6
System A has exactly one solution.
The two equations have the same slope (-8), but different y-intercepts (-6 and 6). This means they are two parallel lines that never intersect, except at one point: the one point where they have the same y-coordinate (in this case, y = -8x - 6 = -8x + 6 at x = 1).
Systems B, C, and D either have infinitely many solutions or no solution.
What is the solution to the system of equations graphed below?
On a coordinate plane, 2 lines intersect at (2, negative 3).
(2, negative 3)
(Negative 3, 2)
(Negative 2, 3)
(3, negative 2)
The solution to the system of equations graphed below is (2, -3).
Since the two lines intersect at (2, -3), this point satisfies both equations in the system. Therefore, (2, -3) is the solution to the system.
The system of equations y = negative three-fourths x minus 1 and y = 2 x minus 4 is shown on the graph below.
On a coordinate plane, a line goes through (negative 2, 0.5) and (0, negative 1) and another goes through (0.5, negative 2) and (2, 0).
What is a reasonable estimate for the solution?
(1.1, –1.9)
(–1.9, 1.1)
(2.1, –3.9)
(–3.9, 2.1)
To find the solution to the system of equations y = -3/4 x - 1 and y = 2x - 4, we need to find the point where these two lines intersect.
From the graph, we can estimate that the point of intersection is approximately (1.1, -1.9).
Therefore, the reasonable estimate for the solution is (1.1, -1.9).
The option that represents this answer is: (1.1, –1.9).
The graph for the equation y = 2 x + 4 is shown below.
On a coordinate plane, a line goes through (negative 2, 0) and (0, 4).
If another equation is graphed so that the system has one solution, which equation could that be?
y = 2 x minus 4
y = 2 (x + 2)
y = 2 (x minus 4)
y = x + 4
We can find the equation of the second line by using the two points given: (-2, 0) and (0, 4).
First, we can find the slope of the line using the slope formula:
slope = (y2 - y1) / (x2 - x1)
slope = (4 - 0) / (0 - (-2))
slope = 2
Next, we can use the slope-intercept form of a line (y = mx + b), and plug in one of the points to solve for the y-intercept:
y = mx + b
0 = 2(-2) + b
0 = -4 + b
b = 4
Therefore, the equation of the second line is y = 2x + 4. This is the same as the equation of the first line, so the system has infinitely many solutions.
To make the system have one solution, the second line must have a different slope than 2. The equation that would achieve this is y = x + 4, so the option that represents this answer is: y = x + 4.