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Nonlinear equations in one variable and systems of equations in two variables: SAT Math notes
Solving equations where the variable is squared, under a root, inside absolute value bars or in a denominator, finding where a line meets a curve, and rearranging formulas to isolate one letter.
Study it with worked examples and 7 practice questionsPractise Nonlinear equations in one variable and systems of equations in two variables questions
What this topic tests
This skill is about equations that are not straight-line equations. The variable might be squared, sit under a square root, sit inside absolute value bars, or appear in a denominator. The goal is still to find the values that make the equation true, but now there can be two answers, one answer, or none at all.
Two related jobs show up a lot. Systems questions ask where a line and a curve, usually a parabola, cross each other, and you solve them by substitution. Formula questions hand you a science or money formula and ask you to get one letter by itself. That is the same algebra as solving an equation, just with letters in place of numbers.
Desmos is a big help here. Graph each side of an equation as its own y = ..., and the x-coordinates of the crossing points are the solutions. For a system, graph both equations and click the intersections. Use algebra when the answer must be exact or when the question has an unknown constant like c or q in it.
Key ideas
- Get a quadratic into the form ax2+bx+c=0 before you factor or use the formula.
- Quadratic formula: x=(-b±√(b2-4ac))/2a.
- The discriminant b2-4ac counts the real solutions: positive means two, zero means one, negative means none.
- Square roots give two answers: if x2=64, then x=8 or x=-8.
- For ax2+bx+c=0, the solutions add up to -b/a and multiply to c/a.
- For a system, substitute one equation into the other so that only one variable is left.
- After squaring both sides or clearing a denominator, check every answer in the original equation. Some can be fake (extraneous) solutions.
- An absolute value can never be negative, so an equation like |4x-3|=-9 has no solutions.
The 4 kinds of Nonlinear equations in one variable and systems of equations in two variables question
Sorted from every Nonlinear equations in one variable and systems of equations in two variables question in the site's bank: each kind keeps coming up, and has a method that works every time.
1. Quadratic equations: solve or count solutions
How to spot it. One equation in one variable with an x2 term, or a product of brackets set equal to zero. The question asks for a solution, the sum or product of the solutions, how many solutions there are, or a constant that makes the equation have a certain number of solutions.
How to answer it:
- Move everything to one side so the equation reads ax2+bx+c=0. If it is already a square equal to a number, like (x+5)2=4, just take the square root of both sides and keep both the plus and the minus.
- Try to factor. If it factors, set each factor equal to zero.
- If it does not factor nicely, use the quadratic formula x=(-b±√(b2-4ac))/2a.
- If the question is about how many solutions there are, or says something like 'exactly one solution' or 'no real solutions', work only with the discriminant b2-4ac. Set it equal to zero, greater than zero or less than zero, then solve for the unknown constant.
- Desmos shortcut: graph y = the left side and y = the right side and read off where they cross. The number of crossings is the number of real solutions.
The trap. Getting a, b or c wrong. The equation must equal zero first, and the signs come along: in x2-x-1=0, b is -1, so -b is +1. Also, when you take a square root, don't drop the negative answer.
2. A line meets a curve (nonlinear systems)
How to spot it. Two equations, usually a line and a parabola (sometimes a circle, or a graph instead of equations). The question asks for a solution (x,y), an intersection point, how many times the graphs cross, or a constant that makes them touch at exactly one point.
How to answer it:
- Get y by itself in the simpler equation, which is usually the line.
- Substitute that expression for y in the other equation, so you have one equation with only x in it.
- Rearrange to ax2+bx+c=0 and solve by factoring or the formula. Each x value gives one intersection point. Plug it back into the line to get y if you need it.
- If the question is about how many intersections there are, use the discriminant of that combined equation. A horizontal line y=k touches a parabola exactly once only when it goes through the vertex.
- Desmos shortcut: type both equations and click the crossing points. For an unknown constant, let Desmos make a slider and drag it until the graphs touch the way the question describes.
The trap. Stopping halfway. Many questions want y, or something like xy or y1+y2, so after you find x, reread the question and finish the job. Also, choices often include a point that works in only one of the two equations, so check your point in both.
3. Solving a formula for one variable
How to spot it. A formula with several letters, often from science or money, and a question like 'Which of the following expresses T in terms of v?' All four choices are rearrangements of the formula.
How to answer it:
- Find the letter you need to get alone. Treat every other letter as if it were just a number.
- If that letter is in a denominator, multiply both sides by the denominator first to bring it up.
- Undo what is around the letter in reverse order: first add or subtract the terms that are separate from it, then multiply or divide.
- If the letter is squared, take the square root as the last step.
- If you are unsure, check with easy numbers: choose values for the other letters, work out the original, and see whether your new formula gives the same result.
The trap. Dividing only part of a side. In 2r=9s+t, you must move t first and then divide the whole side by 9, which gives s=(2r-t)/9, not 2/9r-t. Also watch your signs when you undo a subtraction: undoing -40,000 means adding 40,000.
4. Absolute value, radical, and rational equations
How to spot it. The variable is inside absolute value bars, under a square root, or in the denominator of a fraction. The question asks for a solution, the solution set, or how many solutions there are.
How to answer it:
- Isolate the tricky part first: get the absolute value, the square root, or the fraction by itself on one side.
- Absolute value: if |A|=k, solve both A=k and A=-k. If k is negative, there are no solutions.
- Square root: square both sides, then solve the equation you get, which is often a quadratic.
- Fractions: multiply both sides by the denominator (or the common denominator) to clear the fractions, then solve.
- Check every answer in the original equation. Squaring can create fake solutions, and any value that makes a denominator zero is not allowed. Graphing both sides in Desmos shows only the real solutions.
The trap. Keeping a fake solution. After squaring √(2x+6)=x-1, you get both 5 and -1, but -1 makes the right side negative, and a square root can never equal a negative number. In fraction equations, a value like x=-3 that makes a denominator zero must be thrown out.
Practise Nonlinear equations in one variable and systems of equations in two variables
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