Systems of equations
62 flashcards covering Systems of equations for the GMAT Quantitative section.
Systems of equations refer to a set of two or more equations that share the same variables, and the goal is to find values for those variables that satisfy all equations at once. For instance, you might have two equations like 2x + y = 5 and x - y = 1, and you'd solve them together to determine x and y. This concept is essential in algebra because it helps model real-world problems, such as allocating resources or analyzing relationships between quantities.
On the GMAT Quantitative section, systems of equations often appear in word problems or algebraic setups, where you might need to solve for multiple unknowns or interpret graphical intersections. Common traps include making errors in substitution or elimination methods, or overlooking cases with no solution or infinite solutions. Focus on practicing efficient solving techniques and translating word problems accurately to avoid time-wasting mistakes.
A concrete tip: Always verify your solution by substituting values back into the original equations.
Terms (62)
- 01
System of equations
A system of equations is a set of two or more equations with the same variables, and solving it means finding values for the variables that satisfy all equations simultaneously.
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Solution to a system
The solution to a system of equations is the set of values for the variables that make every equation in the system true at the same time.
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Linear equation
A linear equation is an equation that graphs as a straight line, typically in the form ax + by = c, and in a system, it represents a line on a coordinate plane.
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Substitution method
The substitution method solves a system by solving one equation for one variable and substituting that expression into the other equation to solve for the remaining variable.
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Elimination method
The elimination method solves a system by adding or subtracting equations to eliminate one variable, allowing you to solve for the other variable directly.
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Graphing method for systems
The graphing method solves a system by plotting the equations on a graph and finding the point where the lines intersect, which is the solution.
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Intersection point
The intersection point of two lines on a graph is the coordinate where they cross, representing the solution to the system of equations.
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Consistent system
A consistent system of equations has at least one solution, meaning the equations can be satisfied simultaneously.
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Inconsistent system
An inconsistent system of equations has no solution, typically indicated by parallel lines that never intersect.
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Dependent system
A dependent system of equations has infinitely many solutions, occurring when the equations represent the same line.
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Independent system
An independent system of equations has exactly one unique solution, where the lines intersect at a single point.
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Two-variable system
A two-variable system involves equations with two unknowns, usually solved by substitution, elimination, or graphing to find specific values for both.
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Three-variable system
A three-variable system involves equations with three unknowns, often requiring two equations to eliminate variables step by step until one variable is solved.
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Word problem translation
Translating a word problem into a system means identifying the unknowns and setting up equations based on the relationships described in the problem.
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Break-even point
The break-even point in a business context is the point where costs equal revenues, found by solving a system of equations for cost and revenue functions.
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Supply and demand equations
Supply and demand equations model market equilibrium by setting a supply equation equal to a demand equation and solving the system to find the price and quantity.
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Mixture problems
Mixture problems involve combining substances with different concentrations, solved by setting up a system of equations for the amounts and concentrations.
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Age problems
Age problems use a system of equations to represent relationships between people's ages at different times, such as current age and age in future years.
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Distance-rate-time systems
Distance-rate-time systems set up equations based on the formula distance = rate × time for multiple objects or trips, solving for unknowns like speed or time.
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Extraneous solutions
Extraneous solutions are values that satisfy an equation derived from the system but not the original equations, often occurring in nonlinear systems.
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Quadratic systems
Quadratic systems involve at least one quadratic equation, solved by substitution or elimination, and may require factoring or the quadratic formula.
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Multiplying equations
Multiplying equations in a system means scaling one or both equations by constants to make coefficients equal, facilitating elimination of a variable.
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Checking solutions
Checking solutions involves substituting the found values back into the original equations to verify they satisfy all equations in the system.
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No solution case
The no solution case occurs when equations in a system are inconsistent, such as parallel lines, indicating no values satisfy both equations.
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Infinite solutions case
The infinite solutions case happens when equations are dependent, meaning they are essentially the same and any point on the line works.
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Unique solution
A unique solution is a single set of values that satisfies all equations in the system, typically from intersecting lines at one point.
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Systems with fractions
Systems with fractions require clearing denominators by multiplying equations by the least common multiple to simplify solving.
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Systems with decimals
Systems with decimals can be solved by converting decimals to fractions or multiplying equations by powers of 10 to eliminate decimal points.
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Integer solutions
Integer solutions are cases where the values that satisfy the system are whole numbers, often checked after solving.
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Common errors in substitution
Common errors in substitution include forgetting to substitute correctly or making algebraic mistakes, which can lead to incorrect solutions.
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When to use elimination
Use elimination when equations have coefficients that can be easily made equal, especially if variables have the same or opposite coefficients.
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Graphically solving inequalities
Graphically solving inequalities in a system involves shading regions on a graph where both inequalities hold true, finding the feasible area.
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Feasible region
The feasible region is the area on a graph that satisfies all inequalities in a system, used to identify possible solutions.
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Constraints in word problems
Constraints in word problems are the conditions or limitations given, translated into equations or inequalities for the system.
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Objective function
The objective function in a system represents what you want to maximize or minimize, subject to the constraints of the equations.
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Parametric equations in systems
Parametric equations in systems express variables in terms of a parameter, allowing you to solve for relationships rather than specific values.
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Nonlinear systems
Nonlinear systems include equations that are not straight lines, like circles or parabolas, and require methods like substitution to solve.
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Systems with absolute values
Systems with absolute values involve equations where variables are inside absolute value signs, requiring consideration of different cases for each.
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Balancing equations
Balancing equations in a system means ensuring that the equations are set up correctly to reflect the problem's conditions before solving.
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Overdetermined systems
Overdetermined systems have more equations than variables, often leading to no solution if the equations are inconsistent.
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Underdetermined systems
Underdetermined systems have fewer equations than variables, potentially leading to infinite solutions.
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Example: Simple substitution
For the system x + y = 5 and x = 2, substitute x = 2 into the first equation to get 2 + y = 5, so y = 3, giving the solution (2, 3).
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Example: Elimination with addition
For the system x + y = 7 and x - y = 1, add the equations to eliminate y, resulting in 2x = 8, so x = 4, then y = 3.
Equations: x + y = 7, x - y = 1; Solution: (4, 3)
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Example: Graphing two lines
For y = 2x + 1 and y = -x + 4, graph both lines; they intersect at (1, 3), which is the solution.
Lines intersect at x = 1, y = 3
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Example: Inconsistent system
For x + y = 3 and x + y = 5, the lines are parallel and never intersect, so there is no solution.
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Example: Dependent system
For 2x + y = 4 and 4x + 2y = 8, the second is a multiple of the first, so there are infinitely many solutions.
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Example: Three-variable system
For x + y + z = 6, x - y + z = 2, and 2x + z = 5, solve step by step to find x = 1, y = 2, z = 3.
Solution: x=1, y=2, z=3
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Example: Mixture problem
To mix 10 liters of a 20% solution with a 50% solution to get 30% concentration, solve the system: 0.2x + 0.5y = 0.3( x+y ) and x + y = 10, yielding x=5, y=5.
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Example: Age problem
If Alice is twice as old as Bob and in 5 years she will be 1.5 times his age, solve: A = 2B and A + 5 = 1.5(B + 5), giving A=20, B=10.
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Example: Break-even analysis
For costs C = 100 + 5x and revenue R = 10x, set C = R to get 100 + 5x = 10x, so x=20 units at break-even.
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Strategy for word problems
For word problems, define variables for unknowns, write equations based on given relationships, and solve the system systematically.
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Avoiding algebraic mistakes
To avoid algebraic mistakes in systems, double-check each step, such as substitution or addition, and verify the final solution.
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Using parameters
In underdetermined systems, express solutions in terms of a parameter, like letting one variable be t and solving for others.
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Rationalizing denominators
When solving systems with fractions, rationalize denominators in intermediate steps to simplify expressions and reduce errors.
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Interpreting graphical solutions
Interpreting graphical solutions means identifying the intersection point and ensuring it fits the problem's context.
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Systems in business contexts
In business, systems model scenarios like profit maximization, where equations represent costs, revenues, and constraints.
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Non-integer solutions
Non-integer solutions occur when variables in a system resolve to fractions or decimals, which may still be valid depending on the problem.
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Estimating solutions
Estimating solutions involves approximating values graphically or algebraically when exact solving is complex.
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Error in variable definition
An error in variable definition happens when variables don't accurately represent the unknowns, leading to incorrect equations.
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Scaling equations
Scaling equations means multiplying an equation by a constant to align coefficients, useful in elimination.
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Back-substitution
Back-substitution is using a solved variable to plug back into previous equations in a multi-variable system.
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Feasibility checks
Feasibility checks ensure that solutions make sense in the real-world context, like positive quantities in mixture problems.