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This week in math, we learned about the system of equations. We do this by graphing, inspection, and using algebra and elimination. the equations 3x +y =11 and 2x+3y=12 considered at the same time are systems of equations. The solution to the system of equations is x=3 and y=2. This is because x being 3 and y being 2 satisfies the equation. We also learned the method of substitution and the method of inspection.

Consider the equation x-2y=3 and x+y=0.
Write them in slope-intercept form.
The slope intercept form is y=1/2x-3/2. The other one is y=-x.
Create a table of values for each equation. Plot on a graph.
x|y
-3|-3
-1|-2
1|-1
3|0
5|1

x|y
-4|4
-2|2
0|0
2|-2
4|-4

In the graph in the image below, they intersect at (1,-1).
We can verify the solution by substituting 1 and -1 for the x and y values.
x-2y=3
1-2(-1)=3
1-(-2)=3
3=3.

x+y=0.
1+-1=0.
0=0. Next, we learned about solving by trying different values in our brains. This is the method of inspection. This method is where you just have to mentally try different values with the variables.
Method of Substitution:

When the equation is too hard to solve through inspection, use this method.
1) Choose the easiest equation and express one variable in terms of the other.
2) Substitute the expression from step 1 into the other equation.
3) Solve the single variable equation.
4)  Substitute the equation from step 3 into the equation in step 1 to find the value of the other variable.

Consider the two equations x-2y=10 and x+5y +4 =0.

1)  X= 2Y+10
2) 2Y+10+5Y+4=0
3) 7Y+14=0
4) 7Y=-14
5) Y= -14/7
6) Y= -2
7) Verify

2(-2)+x=10