Mathematics
Grade 10
15 min
Find the number of solutions to a system of equations by graphing
Find the number of solutions to a system of equations by graphing
What you'll learn
- Calculate the sale price of an item when given the original price and the percentage discount with 80% accuracy.
- Explain how to find the sale price of an item in your own words, including the steps involved, to a partner.
- Solve word problems involving sale prices and discounts using mathematical operations like multiplication and subtraction and show your work.
- Identify the original price, discount percentage, and sale price in a given word problem or scenario.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Graph linear equations from both slope-intercept and standard forms.
Visually identify the point of intersection of two graphed lines.
Determine if a system of linear equations has one solution, no solution, or infinitely many solutions by analyzing its graph.
Classify a system as consistent or inconsistent based on its graphical representation.
Relate the slopes and y-intercepts of two linear equations to the number of solutions in the system.
Explain the geometric meaning of each solution type: intersecting, parallel, and coincident lines.
Imagine two rockets launching into space along straight paths. Will their paths cross, and if so, where? 🚀 Graphing helps us visualize and solve problems just like this!
In this tutorial, you will learn how to use g...
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Key Concepts & Vocabulary
TermDefinitionExample
System of Linear EquationsTwo or more linear equations that share the same variables. We consider them together to find a common solution.The equations y = 2x + 1 and y = -x + 7 form a system.
Solution to a SystemAn ordered pair (x, y) that makes all equations in the system true. Geometrically, it is the point where the graphs of the equations intersect.For the system y = x + 2 and y = -x + 4, the ordered pair (1, 3) is the solution because 3 = 1 + 2 and 3 = -1 + 4 are both true.
Intersecting LinesLines that cross at exactly one point. This type of graph represents a system with one unique solution.The graphs of y = 2x and y = -2x intersect at the origin (0,0).
Parallel LinesLines in the same plane that have the same slope but different y-intercepts. They never inter...
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Core Formulas
One Solution (Consistent, Independent System)
Lines intersect at a single point.
This occurs when the slopes of the two lines are different. The y-intercepts can be the same or different. For lines y = m_1x + b_1 and y = m_2x + b_2, this means m_1 \neq m_2.
No Solution (Inconsistent System)
Lines are parallel.
This occurs when the lines have the same slope but different y-intercepts. They will never cross. For lines y = m_1x + b_1 and y = m_2x + b_2, this means m_1 = m_2 and b_1 \neq b_2.
Infinitely Many Solutions (Consistent, Dependent System)
Lines are coincident (the same line).
This occurs when the lines have the same slope and the same y-intercept. They overlap at every point. For lines y = m_1x + b_1 and y = m_2x + b_2, this means m_1 = m_2 and b_1 = b_2.
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Challenging
For the system 5x - 2y = 10 and kx - 4y = 25 to be inconsistent, what must be the value of k?
A.5
B.10
C.-10
D.-5
Challenging
The system of equations Ax + 3y = C and 2x + 6y = D has infinitely many solutions. What is the relationship between A and C, and D?
A.= 1 and C = D
B.= 4 and C = 2D
C.= 1 and 2C = D
D.= 4 and C = D
Challenging
Two cell phone plans are modeled by the equations C = 0.10m + 40 and C = 0.15m + 30, where C is the total cost and m is the number of minutes used. How many solutions does this system have, and what does the solution represent?
A.No solution; the costs are never the same.
B.Infinitely many solutions; the plans are identical.
C.One solution; the number of minutes where the costs are equal.
D.Two solutions; there are two times the costs are equal.
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What grade level is "Find the number of solutions to a system of equations by graphing"?
Find the number of solutions to a system of equations by graphing is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Find the number of solutions to a system of equations by graphing?
You'll be able to: Calculate the sale price of an item when given the original price and the percentage discount with 80% accuracy; Explain how to find the sale price of an item in your own words, including the steps involved, to a partner; Solve….
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How many practice questions are included with Find the number of solutions to a system of equations by graphing?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.