SAT System of Equations

What is the procedure for solving a NO SOLUTION system of equations SAT question?

1. Convert equations to slope-intercept form 2. Set slopes equal to each other, cancelling out x 3. Solve for k, or whatever constant is in the question

What is the procedure for solving a system of equations that involves at least one quadratic equation?

1. Isolate y for both equations 2. Set equal to each other 3. Set equal to 0 4. Factor and solve for x.

What is the procedure for solving an INFINITELY MANY SOLUTIONS systems of equations SAT question?

1. Make sure the variables of the equations align vertically 2. Identify the factor by which the equations are related 3. Use factor to solve for constant

What are the two methods of finding the solution in a system of equations?

1. Substitution 2. Elimination

What is a system of equations?

2 or more equations graphed on the same coordinate plane; they can be linear, quadratic, or even circles

Solve: 𝑎𝑥 + 𝑏𝑦 = 12 2𝑥 + 8𝑦 = 60 In the system of equations above, a and b are constants. If the system has infinitely many solutions, what is the value of a/b?

Answer: 1/4 or 0.25

Solve: 𝑘𝑥 − 3𝑦 = 4 4𝑥 − 5𝑦 = 7 In the system of equations above, k is a constant and x and y are variables. For what value of k will the system of equations have no solution?

Answer: 12/5

Solve: 3x +2y=60 2x+3y=80 What is 5x + 5y?

Answer: 140

Example: If the equation 5x+3=ax+ab has no solution, which of the following must be true? a =5 b = 3/5 b ≠3/5 A. I only B. I and II only C. I and III only D. None of the above

Answer: C

Solve: The function f and g, defined by 𝑓(𝑥) = 8𝑥^2 − 2 and 𝑔(𝑥) = −8𝑥^2 + 2, are graphed in the same xy-plane. The graphs of f and g intersect at the points (k,0) and (-k,0). What is the value of k?

Answer: k = 1/2

Special cases: sometimes a system of equations problem will not ask for the value of x or y, but for the value of a seemingly whole new equation.

Solving for each individual variable is a waste of time. Instead, try to manipulate the systems in some way so that when added, the two equations equal what the question is asking for.

What does it mean when the system has NO solution?

The lines do not intersect therefore they are parallel and have the same slope but different y-intercepts

What does it mean when the system has INFINITELY MANY solutions?

The lines share infinitely many points in common therefore they must be the same line and the equations, when reduced, will be equivalent

In a system of equations, what does the "solution" mean?

The solution is the coordinate point of intersection between the 2 or more equations