Graph the linear system and estimate the solution. Then check the solution algebraically.
7x+y=-17
3x-10y=24
I have to have 4 labeled points and I'm not getting the fraction thing. Any help would be great!
The first equation has a y-intercept of -17.
The x-intercept is -17/7=-(2+3/7)
=-2.4286 approx.
The second equation has a y-intercept of -2.4, and x-intercept of -2.4/(-0.3)=8.
If you produce the two lines to intersection, the estimated solution would be (-2, -3)
To graph the linear system, we will convert each equation into slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.
For the first equation, 7x + y = -17, we need to isolate y on one side:
y = -7x - 17
For the second equation, 3x - 10y = 24, isolate y on one side:
10y = 3x - 24
y = (3/10)x - (24/10)
y = (3/10)x - 12/5
Now we have both equations in slope-intercept form.
To graph the linear system, we can use the slope-intercept form to find the y-coordinate of several points by substituting different x-values in the equations. Here, we can choose 4 x-values and find the corresponding y-values. Let's choose x = -3, -2, 2, 3:
For x = -3:
Using the first equation: y = -7(-3) - 17 = 4
Using the second equation: y = (3/10)(-3) - 12/5 = -9/5
So, the first point is (-3, 4) and (-3, -9/5).
For x = -2:
Using the first equation: y = -7(-2) - 17 = 3
Using the second equation: y = (3/10)(-2) - 12/5 = -3/5
So, the second point is (-2, 3) and (-2, -3/5).
For x = 2:
Using the first equation: y = -7(2) - 17 = -31
Using the second equation: y = (3/10)(2) - 12/5 = -21/5
So, the third point is (2, -31) and (2, -21/5).
For x = 3:
Using the first equation: y = -7(3) - 17 = -38
Using the second equation: y = (3/10)(3) - 12/5 = -9/5
So, the fourth point is (3, -38) and (3, -9/5).
Now you have four labeled points. Plot these points on a graph and connect them with a straight line. It is important to label the points to differentiate between them.
To algebraically check the solution, substitute one of the points into both equations, and if the left side of the equation equals the right side, the point is a solution.
Let's check the point (-2, 3) in both equations:
For the first equation: 7x + y = -17
7(-2) + 3 = -17
-14 + 3 = -17
-11 = -17 (Both sides are equal, so the point satisfies the first equation)
For the second equation: 3x - 10y = 24
3(-2) - 10(3) = 24
-6 - 30 = 24
-36 = 24 (Both sides are not equal, so the point doesn't satisfy the second equation)
Since the point (-2, 3) doesn't satisfy both equations, it is not a solution to the linear system. You can repeat the above process to check the remaining points and find the solution(s) to the system.