Which description best describes the solution to the following system of equations?
y = −one halfx + 9
y = x + 7
Lines y = −one halfx + 9 and y = x + 7 intersect the x-axis.
Lines y = −one halfx + 9 and y = x + 7 intersect the y-axis.
Line y = −one halfx + 9 intersects the origin.
Line y = −one halfx + 9 intersects line y = x + 7.
geez - ever think of using actual numbers, like oh, say -- 1/2 ??
since neither line is horizontal or vertical, and their slopes are different, all of the statements are true except the third one.
However, only the fourth one has anything to do with the solution -- that is, where they intersect.
correct
To determine the solution to the system of equations, we need to find the point where the two lines intersect.
First, let's equate the two equations:
−one halfx + 9 = x + 7
To simplify this equation, let's get rid of the fractions by multiplying both sides by 2:
-2(one halfx + 9) = 2(x + 7)
-2/2x -18 = 2x + 14
-x -18 = 2x + 14
Now, we can solve for x by combining like terms:
-x -2x = 14 + 18
-3x = 32
To isolate x, we need to divide both sides by -3:
-3x/-3 = 32/-3
x = -10.67 (rounded to two decimal places)
Now that we have the value of x, we can substitute it back into one of the original equations to find the corresponding value of y. Let's use the first equation:
y = -one half(-10.67) + 9
y = 5.33 + 9
y = 14.33 (rounded to two decimal places)
So, the point of intersection is approximately (-10.67, 14.33).
From the answer choices, we can see that "Line y = −one halfx + 9 intersects line y = x + 7" is the correct description of the solution to the system of equations.
To determine the solution to the given system of equations, we can analyze the equations and their relationship to each other.
The given system of equations is:
y = -1/2x + 9
y = x + 7
To find the solution to the system of equations, we need to identify the point(s) at which both equations intersect. The point(s) of intersection represent the coordinates (x, y) that satisfy both equations simultaneously.
To solve the system, we can set the two equations equal to each other and solve for x:
-1/2x + 9 = x + 7
We then proceed to solve for x, by first combining like terms:
-1/2x - x = 7 - 9
-3/2x = -2
Next, we multiply both sides of the equation by -2/3 to isolate x:
x = (-2)(-2/3)
x = 4/3
x = 1.33 (rounded to two decimal places)
Once we have found the value of x, we substitute it back into either of the given equations to find the corresponding value of y:
y = -(1/2)(1.33) + 9
y = -0.665 + 9
y = 8.335 (rounded to three decimal places)
So, the solution to the system of equations is (x, y) = (1.33, 8.335).
Now, let's analyze the given options:
1. Lines y = −1/2x + 9 and y = x + 7 intersect the x-axis.
To determine if this option is correct, we need to check if the equations intersect the x-axis. In this case, the y-coordinate must be zero. However, the solution we obtained does not have a zero y-coordinate. Therefore, this option is incorrect.
2. Lines y = −1/2x + 9 and y = x + 7 intersect the y-axis.
Similarly, to check if this option is correct, we need to verify if the equations intersect the y-axis, meaning the x-coordinate is zero. However, the solution we found does not have a zero x-coordinate. Hence, this option is also incorrect.
3. Line y = −1/2x + 9 intersects the origin.
To check if this option is correct, we need to determine if the coordinates (0, 0) satisfy the equation y = -1/2x + 9. Substituting x = 0 and y = 0 into the equation gives:
0 = -1/2(0) + 9
0 = 9
Since this equation is not true, the line does not pass through the origin. Therefore, this option is incorrect.
4. Line y = -1/2x + 9 intersects line y = x + 7.
To check if this option is correct, we need to determine if the two given lines intersect at a point. The solution we found for the system of equations (x, y) = (1.33, 8.335) satisfies both equations simultaneously. Therefore, this option is correct.
Hence, the correct description of the solution to the given system of equations is: "Line y = -1/2x + 9 intersects line y = x + 7."