Mathematics
Grade 11
15 min
Is (x, y) a solution to the system of equations?
Is (x, y) a solution to the system of equations?
What you'll learn
- Identify that any number divided by 1 equals that same number with 100% accuracy on a worksheet with at least 5 problems.
- Solve division problems where the divisor is 1, correctly answering at least 4 out of 5 problems on a quiz.
- Explain, using words or pictures, why any number divided by 1 equals that same number, in a written response of at least two sentences.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define what constitutes a solution to a system of two or more equations.
Accurately substitute the coordinates of an ordered pair (x, y) into various types of equations, including linear, quadratic, and conic sections.
Verify whether a given ordered pair is a solution to a system by checking it in all equations of the system.
Distinguish between an ordered pair that is a solution to one equation versus a solution to the entire system.
Interpret the meaning of obtaining a true statement (e.g., 5 = 5) versus a false statement (e.g., 5 = 7) after substitution.
Articulate why a single point must satisfy every equation in a system to be considered a solution.
Imagine two satellites orbiting Earth on different paths. How can we confirm the exact point in space...
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Key Concepts & Vocabulary
TermDefinitionExample
System of EquationsA set or collection of two or more equations with the same set of unknown variables.The set of equations y = 2x + 1 and y = -x + 4 forms a system.
Ordered PairA pair of numbers, written as (x, y), that represents a specific point on a coordinate plane. The first value is the x-coordinate, and the second is the y-coordinate.(3, -2) represents a point 3 units to the right and 2 units down from the origin.
Solution to a System of EquationsAn ordered pair (x, y) that makes every equation in the system a true statement when the values of x and y are substituted into them. Geometrically, it is the point of intersection of the graphs of the equations.For the system y = x + 2 and y = -x + 4, the ordered pair (1, 3) is the solution because 3 = 1 + 2 (True)...
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Core Formulas
The Solution Verification Principle
An ordered pair (a, b) is a solution to a system of equations if and only if substituting x=a and y=b into *every* equation in the system results in a true statement for all of them.
Use this principle as the definitive test. If the ordered pair fails to produce a true statement for even one of the equations, it is not a solution to the system.
System of Two Equations Condition
For a system with equations f(x, y) = c_1 and g(x, y) = c_2, the point (a, b) is a solution if f(a, b) = c_1 AND g(a, b) = c_2.
This is the formal mathematical representation of the verification principle. The 'AND' is critical; both conditions must be met simultaneously.
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Challenging
A student concludes that (4, 1) is a solution to the system: x² - y² = 15 and x² + y² = 25. Their work shows: "Check 1: x² - y² = 15 → 4² - 1² = 16 - 1 = 15. It works. So, (4, 1) is the solution." What is the fundamental flaw in their reasoning?
A.They made an arithmetic error in the first check.
B.They failed to verify the point in the second equation of the system.
C.They swapped the x and y coordinates in their substitution.
D.They should have graphed the equations instead of using algebra.
Challenging
The point (-2, 3) is confirmed to be a solution to the system of equations: y = -2x - 1 and x² + ky² = 40. What must be the value of the parameter k?
A.k = 2
B.k = -4
C.k = 4
D.k = 36
Challenging
Is the point with coordinates (4cos(π), 5sin(π/2)) a solution to the system x² + y² = 41 and y - x = 9?
A.Yes, it is a solution to the system.
B.No, because it fails to satisfy the first equation.
C.No, because it fails to satisfy the second equation.
D.No, because trigonometric values cannot be coordinates.
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What grade level is "Is (x, y) a solution to the system of equations?"?
Is (x, y) a solution to the system of equations? is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Is (x, y) a solution to the system of equations??
You'll be able to: Identify that any number divided by 1 equals that same number with 100% accuracy on a worksheet with at least 5 problems; Solve division problems where the divisor is 1, correctly answering at least 4 out of 5 problems on a….
Is "Is (x, y) a solution to the system of equations?" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Is (x, y) a solution to the system of equations??
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.