Mathematics
Grade 12
15 min
Solve a system of equations by graphing (Tutorial only)
Solve a system of equations by graphing (Tutorial only)
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Identify the solution(s) to a system of equations as the point(s) of intersection on a graph.
Accurately graph functions from various families, including linear, quadratic, cubic, and trigonometric.
Solve systems involving both linear and non-linear functions by graphing.
Determine the number of real solutions (zero, one, two, or more) for a system by analyzing the intersection of their graphs.
Estimate solutions from a hand-drawn graph and find precise solutions using graphing technology.
Verify a graphical solution by substituting the coordinates back into the original system of equations.
Where does a satellite's orbital path cross the trajectory of a comet? 🛰️ Solving this is like finding the intersection point in a cosmic system of equations!
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Key Concepts & Vocabulary
TermDefinitionExample
System of EquationsA collection of two or more equations that share the same set of variables.The pair of equations y = x^2 - 4 and y = -x - 2 forms a system.
Solution to a SystemAn ordered pair (or set of values) that simultaneously satisfies every equation in the system. Geometrically, it is a point of intersection of all the graphs.For the system y = x+1 and y = x^2-1, the point (2, 3) is a solution because 3 = 2+1 and 3 = 2^2-1 are both true.
Point of IntersectionThe specific point (x, y) where the graphs of two or more functions cross or touch each other.The graphs of y = 2x and y = -x + 3 intersect at the point (1, 2).
Family of FunctionsA group of functions that share a common structural form and graphical characteristics. The specific graph is determined by p...
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Core Formulas
The Graphical Solution Principle
The set of real solutions to a system of equations corresponds to the set of all points of intersection of their graphs.
To solve a system graphically, you graph each equation on the same coordinate plane and find the coordinates of the points where the graphs cross. Each point of intersection is a solution.
The Verification Principle
A point (x_0, y_0) is a solution to the system \( y = f(x) \) and \( y = g(x) \) if and only if \( y_0 = f(x_0) \) AND \( y_0 = g(x_0) \).
After finding a potential solution (x₀, y₀) from a graph, you must substitute x₀ and y₀ into *all* original equations to confirm that they create true statements for every equation in the system.
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Challenging
For the system y = |x^2 - 4| and y = k, for which value of k will the system have exactly three real solutions?
A.k = 0
B.k = 2
C.k = 4
D.k = 5
Challenging
Consider the system y = e^x and y = x + k. There is exactly one value of k for which the system has exactly one solution. At this point, the line y = x + k is tangent to the curve y = e^x. What is the relationship between the two functions at this point of tangency (x₀, y₀)?
A.Their y-intercepts are equal.
B.Their second derivatives are equal.
C.Their values and their first derivatives are equal.
D.The line is the inverse of the exponential function.
Challenging
For the system y = a*cos(x) and y = x^2 + 1, where a > 1, how does increasing the value of the parameter 'a' affect the number of solutions?
A.The number of solutions increases.
B.The number of solutions decreases.
C.The number of solutions remains the same.
D.The effect depends on the initial value of 'a'.
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