Where do the graphs of these equations intersect?
3x+y=4 ((0,4) ???
X+y=2 (0,2) ???
Write an equation that results from adding the 2 equations in the above problem.
Does this equation also pass through (1,1) ??
3x+y=4
X+y=2
subtract the two equations:
2x=3 ,or x=1. If x=1, then y=1 (second equation), so they intersect at (1,1).
3 x + y = 4
1 x + y = 2
----------------- subtract
2 x + 0 = 2
x = 1
1 + y = 2
y = 1 so the two lines intersect at (1 , 1)
now add the original two lines
4 x + 2 y = 6
if y = 1
4x + 2 = 6
4 x = 4
x = 1
so yes , the sum goes through (1,1)
Your first question probably was:
Where do the graphs of these equations intersect the x-axis ?
the answers you gave are correct
To find the intersection point of two equations, you can solve the system of equations by either substitution or elimination method.
1. Substitution Method:
a. Solve one equation for one variable in terms of the other variable.
Let's solve the second equation, "x + y = 2," for x:
x = 2 - y
b. Substitute the value of x in the other equation.
Plug x into the first equation:
3(2 - y) + y = 4
Simplify:
6 - 3y + y = 4
-2y = -2
Divide both sides by -2:
y = 1
c. Substitute the value of y back into either of the original equations to find x.
Plug y = 1 into the second equation:
x + 1 = 2
x = 2 - 1
x = 1
Therefore, the intersection point is (1, 1).
2. Elimination Method:
a. Multiply both sides of the second equation by 3 to make the coefficients of y in both equations the same.
3(x + y) = 3(2)
3x + 3y = 6
b. Subtract the first equation from the modified second equation to eliminate y.
(3x + 3y) - (3x + y) = 6 - 4
Simplify:
3y - y = 2
2y = 2
Divide both sides by 2:
y = 1
c. Substitute the value of y back into either of the original equations to find x.
Plug y = 1 into the second equation:
x + 1 = 2
x = 2 - 1
x = 1
As a result, the intersection point is (1, 1).
Now, let's write an equation that results from adding the two equations:
3x + y = 4 (Equation 1)
+ x + y = 2 (Equation 2)
--------------
4x + 2y = 6
So, the equation obtained after adding the two given equations is 4x + 2y = 6.
Lastly, to determine if the equation 4x + 2y = 6 passes through the point (1, 1), we can substitute the values of x and y into the equation and check if it holds true.
Substitute x = 1, y = 1 into the equation:
4(1) + 2(1) = 6
4 + 2 = 6
6 = 6
Since the equation holds true when substituting (1, 1), the point (1, 1) does lie on the graph of the equation 4x + 2y = 6.