How to solve Passport to Advanced Math Problem from SAT Math Easily
Passport to Advanced Math section in SAT Math questions tests the skill to construct a mathematical expression like a quadratic or exponential function or equation from the information given in a problem. It also may involve solving a quadratic equation, performing mathematical operations on polynomial expressions, understanding non linear relationship and functional notations etc.
Sample problem from SAT Math on Passport to Advanced Math solved here
A system of three equations and their graphs in the xy-plane are shown above. How many solutions does the system have?
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Well, what do you think. The concept here what does it mean when an equation is represented on a graph. Here three equations are on a graph. They have given the equations as well. Now the questions is how many solutions does the system have?
The concept being tested is how to find the solution of a system of equations.
Well an equation represent a relationship between variables.
So the first equation is
What it means if I know either x or y- say for argument sake if know a value of x say 1. Then knowing this equation I can find the value of y. How?
Subtracting x2 on both sides
Y2 = 5 – x2
Therefore, y2= 5-(1)2
Y2 = 4
Taking square root y = ±2.
What does this mean, it means for x=1, the value of y that satisfies x2+y2=5 are +1, and -1. What does this mean in a graph, it means that if you the points (1,-1) and (1,1) both lie on the graph x2+y2 =5.
Note that the circle represented in figure in the question above is the graph of the first function: x2+y2=5
The second equation y =x2-3 is represented by the curve and the third equation x-y=1 is presented by a line.
Note that all these figures i.e. the circle, the curve and the line intersect distinctly at two points., once somewhere at (-1,-2) and the other at (2,1). What does that mean. It means that at these two points, if you plug in the x and y value in the three equations, the equations are satisfied. What doe s this mean
(-1)2 + (-2)2 = 1 +4 =5 = RHS of equation 1
-2 = (-1)2 -3 =1-3 = LHS of equation 2
-1-(-2) = -1+2 = 1 = RHS of equation 3.
If you plug in (2,1) in all three equations, you will see that the equations are satisfied.
Thus, thereare 2 solutions to the systems of equation.
Which means if plotted on a graph, if the lines, or shapes intersect, it represents a solution to those system of equations.
Hope you understand the logic of solving these equations.
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