Systems word problems
56 flashcards covering Systems word problems for the SAT Math section.
Systems of equations word problems involve translating real-world scenarios into a set of two or more equations to solve for unknown variables. For example, you might need to determine how many items of different types to buy or mix to meet specific conditions, like cost or quantity limits. This topic builds problem-solving skills by showing how multiple relationships can intersect, making it essential for applying algebra to everyday situations.
On the SAT Math section, these problems typically appear as multiple-choice or grid-in questions, often in the context of linear equations or simple graphs. They test your ability to accurately set up equations from word descriptions and solve them efficiently, with common traps including misinterpreting key details or algebraic mistakes. Focus on practicing clear translation from words to math and double-checking your work to avoid errors.
Remember to label your variables clearly from the start.
Terms (56)
- 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 that satisfy all equations simultaneously.
- 02
Linear system
A linear system consists of linear equations, where each equation graphs as a straight line, and solutions are found by determining intersection points.
- 03
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.
- 04
Elimination method
The elimination method adds or subtracts equations in a system to eliminate one variable, allowing you to solve for the other variable and then back-substitute.
- 05
Graphing systems
Graphing systems involves plotting the equations on a coordinate plane to find their intersection point, which represents the solution.
- 06
Solution to a system
The solution to a system is the set of values for the variables that make all equations true at the same time, often a single point for two linear equations.
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Consistent system
A consistent system has at least one solution, meaning the equations intersect at one or more points.
- 08
Inconsistent system
An inconsistent system has no solution, occurring when equations represent parallel lines that never intersect.
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Dependent system
A dependent system has infinitely many solutions, happening when equations are essentially the same line.
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Independent system
An independent system has exactly one solution, where the equations intersect at a single point.
- 11
Mixture problems
Mixture problems involve combining two or more substances with different concentrations to form a new mixture, typically solved using a system of equations for amounts and concentrations.
- 12
Rate problems
Rate problems deal with speeds or rates of change, using systems to relate distance, rate, and time for multiple objects or scenarios.
- 13
Work rate problems
Work rate problems calculate how long it takes for individuals or machines to complete a task together, using systems to combine their rates.
- 14
Age problems
Age problems use systems to relate the current ages of people and their ages at future or past times, based on given relationships.
- 15
Distance problems
Distance problems involve objects traveling at different speeds, using systems to find when or where they meet based on distance formulas.
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Break-even point
The break-even point is the level of production or sales where total costs equal total revenue, found by solving a system of cost and revenue equations.
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Extraneous solutions
Extraneous solutions are values that satisfy the equations but do not fit the original problem's context, often checked by plugging back into the problem.
- 18
Checking solutions
Checking solutions involves substituting the found values back into the original equations to verify they work and ensure no errors.
- 19
Setting up equations from words
Setting up equations from words requires identifying variables for unknowns and translating phrases into mathematical equations based on the problem's conditions.
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Variables in word problems
Variables in word problems represent unknown quantities, and you must define them clearly at the start to set up the system accurately.
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Systems with three variables
Systems with three variables require three equations to solve for all unknowns, often using elimination or substitution in a step-by-step manner.
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Inequalities in systems
Inequalities in systems represent constraints, and solutions are regions on a graph that satisfy all inequalities simultaneously.
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Feasible region
The feasible region is the area on a graph that satisfies all inequalities in a system, and optimal solutions are often at its vertices.
- 24
Corner points
Corner points are the vertices of the feasible region in a system of inequalities, where maximum or minimum values typically occur.
- 25
Strategy for substitution
Use substitution when one equation is already solved for a variable or when it's easy to solve for one, as it simplifies the system step by step.
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Strategy for elimination
Use elimination when coefficients of variables can be made the same to cancel out, making it efficient for systems with multiple equations.
- 27
Common errors in substitution
Common errors in substitution include algebraic mistakes when substituting expressions or forgetting to solve for both variables.
- 28
Common errors in elimination
Common errors in elimination involve incorrect addition or subtraction of equations, leading to wrong coefficients or lost variables.
- 29
Interpreting negative solutions
Interpreting negative solutions requires checking if they make sense in the context, as negative values for quantities like age or distance are often invalid.
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Real-world constraints
Real-world constraints in systems limit solutions to positive or integer values, ensuring they fit the problem's practical scenario.
- 31
Cost and revenue equations
Cost and revenue equations in systems relate fixed and variable costs to sales, helping find profit or break-even points.
- 32
Supply and demand
Supply and demand in systems use equations to find equilibrium price and quantity where supply equals demand.
- 33
Parallel lines in systems
Parallel lines in systems indicate no solution if they have the same slope but different y-intercepts.
- 34
Perpendicular lines in systems
Perpendicular lines in systems have slopes that are negative reciprocals, intersecting at a right angle.
- 35
Ratio and proportion in systems
Ratio and proportion in systems set up equations based on given ratios to solve for unknown quantities.
- 36
Alligation method
The alligation method simplifies mixture problems by using weighted averages to find ratios without full systems, but can be part of a system setup.
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Harmonic mean in rates
The harmonic mean calculates average rates in systems, especially for equal distances traveled at different speeds.
- 38
Work rates addition
Work rates addition combines individual rates in a system to find the combined rate for tasks done together.
- 39
Relative speed
Relative speed in systems is the difference in speeds of moving objects, used to determine meeting times.
- 40
Coin problems
Coin problems use systems to relate the number and value of coins to find how many of each type satisfy the total.
- 41
Digit problems
Digit problems set up systems where digits of a number are variables, relating them through equations based on the number's properties.
- 42
Number problems with systems
Number problems with systems involve finding pairs or triples of numbers that satisfy given conditions, like sums or products.
- 43
Investment problems
Investment problems use systems to allocate amounts into different investments with varying interest rates to meet total return goals.
- 44
Simple interest in systems
Simple interest in systems calculates interest earned on investments, using formulas to set up equations for total amounts.
- 45
Population growth systems
Population growth systems model changes over time with equations, often linear for short periods, to predict future populations.
- 46
Absolute value in systems
Absolute value in systems creates piecewise equations, requiring consideration of different cases for positive and negative values.
- 47
Quadratic systems
Quadratic systems involve at least one quadratic equation, solved by substitution or elimination, with solutions as intersection points.
- 48
Word problems with percentages
Word problems with percentages use systems to relate parts, wholes, and percentages in scenarios like discounts or mixtures.
- 49
When to use graphing
Use graphing for systems when visual estimation is helpful, such as for inequalities or when equations are simple to plot.
- 50
Units in systems
Units in systems must be consistent across equations to ensure solutions are meaningful and avoid errors in interpretation.
- 51
Overcounting in systems
Overcounting in systems occurs when variables represent overlapping quantities, leading to incorrect setups.
- 52
Undercounting in systems
Undercounting in systems happens when not all variables or conditions are included, resulting in incomplete solutions.
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Example of mixture problem
In a mixture problem, if 10 liters of 20% solution and x liters of 50% solution make 15 liters of 30% solution, solve the system for x.
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Example of rate problem
In a rate problem, if two cars leave at the same time and meet after 2 hours, with speeds of 50 km/h and 60 km/h, set up a system to find the distance.
- 55
Example of work rate
In a work rate problem, if A paints a house in 4 hours and B in 6 hours, their combined rate forms a system to find time together.
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Example of age problem
In an age problem, if Alice is twice as old as Bob now, and in 5 years she'll be 1.5 times his age, solve the system for their current ages.