Equation A : 3x + y = 23
Equation B : 4x - y = 12
Use the system of equations to determine, which of the following statements are true select all that apply
1: The solution to the system of equations is (5,8)
2: (2,-4) is a solution to Equation B
3: (2,-4) is a solution to neither Equation
4: The y-intercept of Equation B is (0,12)
5: The slope of the Equation A is M = -3
Let's solve the system of equations:
Equation A: 3x + y = 23
Equation B: 4x - y = 12
To find the solution to the system, we can use any of the methods such as substitution or elimination. Let's use the elimination method:
Multiply Equation A by 4 and Equation B by 3 to eliminate the y term:
4(3x + y) = 4(23)
3(4x - y) = 3(12)
12x + 4y = 92
12x - 3y = 36
Now subtract the second equation from the first equation:
(12x + 4y) - (12x - 3y) = 92 - 36
12x + 4y - 12x + 3y = 56
7y = 56
y = 8
Now substitute the value of y into either of the original equations. Let's use Equation A:
3x + 8 = 23
3x = 23 - 8
3x = 15
x = 5
So, the solution to the system of equations is (5, 8). Therefore, statement 1 is true.
Now let's check the other statements:
2: (2, -4) is a solution to Equation B:
For Equation B: 4x - y = 12
4(2) - (-4) = 12
8 + 4 = 12
12 = 12
Therefore, (2, -4) is a solution to Equation B. So, statement 2 is true.
3: (2, -4) is a solution to neither equation:
Since we just found that (2, -4) is a solution to Equation B, statement 3 is false.
4: The y-intercept of Equation B is (0, 12):
The y-intercept is the value of y when x = 0. Let's substitute x = 0 into Equation B:
4(0) - y = 12
0 - y = 12
-y = 12
y = -12
Therefore, the y-intercept of Equation B is (0, -12). So, statement 4 is false.
5: The slope of Equation A is M = -3:
The slope-intercept form of a linear equation is y = mx + b, where m is the slope.
Equation A can be rewritten as y = -3x + 23, so the slope of Equation A is -3. Therefore, statement 5 is true.
In conclusion, the true statements are 1 and 5.