Find the solution to this system:%0D%0A%0D%0A%0D%0AThe horizontal axis extends from negative 1 to 3. The vertical axis extends from negative 1 to 4.%0D%0AOne line intersects the horizontal axis at negative 1 and the vertical axis at 1.%0D%0AThe second line intersects the horizontal axis at 3 and the vertical axis at 3.%0D%0AThe two lines intersect at the point 1 comma 2.%0D%0A(1 point)%0D%0AResponses%0D%0A%0D%0A(2, 1)%0D%0A(2, 1)%0D%0A%0D%0A(1, –2)%0D%0A(1, –2)%0D%0A%0D%0A(1, 2)%0D%0A(1, 2)%0D%0A%0D%0A(–1, 2)
The correct solution to the system is (1, 2).
To find the solution to the system, we can identify the equations of the two lines based on the given information.
1. The first line intersects the horizontal axis at negative 1 and the vertical axis at 1. This means it passes through the points (-1,0) and (0,1). We can find the equation of this line using the slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.
The slope (m) can be calculated as (change in y)/(change in x) = (1 - 0)/(0 - (-1)) = 1/1 = 1.
Substituting the slope and the point (0,1) into the equation, we get y = 1x + 1, which simplifies to y = x + 1.
2. The second line intersects the horizontal axis at 3 and the vertical axis at 3. This means it passes through the points (3,0) and (0,3). Using the same process as above, we find the equation of this line to be y = -x + 3.
To find the point of intersection of these two lines, we can set their equations equal to each other:
x + 1 = -x + 3
2x = 2
x = 1
Substituting the value of x into either equation, we find:
y = 1 + 1 = 2
Therefore, the point of intersection is (1, 2).
So, the correct response is (1, 2).
To find the solution to this system, we need to determine the equations of the two lines and then solve them simultaneously.
Based on the given information, we know that one line intersects the horizontal axis at -1 and the vertical axis at 1, and the other line intersects the horizontal axis at 3 and the vertical axis at 3. We also know that these two lines intersect at the point (1, 2).
Let's find the equations of these two lines:
1. Line 1:
Since it intersects the horizontal axis at -1 and the vertical axis at 1, we can write two points on this line as (-1, 0) and (0, 1). Using these two points, we can find the slope of the line using the formula:
slope = (y2 - y1) / (x2 - x1)
slope = (1 - 0) / (0 - (-1)) = 1/1 = 1
Now we can use the slope-intercept form of a linear equation (y = mx + b) to find the equation of the line. Substituting one of the given points, let's choose (-1, 0):
0 = 1*(-1) + b
0 = -1 + b
b = 1
So the equation of Line 1 is: y = x + 1
2. Line 2:
Since it intersects the horizontal axis at 3 and the vertical axis at 3, we can write two points on this line as (3, 0) and (0, 3). Using these two points, we can find the slope of the line using the formula:
slope = (y2 - y1) / (x2 - x1)
slope = (3 - 0) / (0 - 3) = 3/(-3) = -1
Now we can use the slope-intercept form of a linear equation (y = mx + b) to find the equation of the line. Substituting one of the given points, let's choose (0, 3):
3 = -1*(0) + b
3 = b
So the equation of Line 2 is: y = -x + 3
To solve these two equations simultaneously, we can substitute one equation into the other:
y = x + 1 (Equation 1)
y = -x + 3 (Equation 2)
Since both equations are equal to y, we can set them equal to each other:
x + 1 = -x + 3
Now let's solve for x:
2x = 2
x = 1
Substituting this value back into either of the original equations, let's use Equation 1:
y = 1 + 1
y = 2
Therefore, the solution to this system of equations is (1, 2). So the correct response from the given options is (1, 2).