Mathematics
Grade 8
15 min
Solve a system of equations by graphing: word problems
Solve a system of equations by graphing: word problems
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1
Introduction & Learning Objectives
Learning Objectives
Translate real-world scenarios into a system of two linear equations.
Accurately graph two linear equations on the same coordinate plane.
Identify the intersection point of two graphed lines.
Interpret the meaning of the intersection point as the solution to a word problem.
Verify the solution of a system of equations by substituting values back into the original equations.
Explain the limitations and advantages of solving systems by graphing.
Ever wonder when two different options, like phone plans or rental car costs, become equal? 🤝 Graphing can help us find that exact moment!
In this lesson, you'll learn how to take real-world situations described in words, turn them into mathematical equations, and then use graphs to find the solution. This sk...
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Key Concepts & Vocabulary
TermDefinitionExample
System of Linear EquationsA set of two or more linear equations that use the same variables. We look for values of the variables that satisfy ALL equations in the system simultaneously.Equation 1: $y = 2x + 3$
Equation 2: $y = -x + 6$
Linear EquationAn equation whose graph is a straight line. It typically involves two variables, usually 'x' and 'y', each raised to the power of one.$y = 3x - 5$ or $2x + y = 7$
Solution to a SystemThe ordered pair (x, y) that makes ALL equations in the system true. Geometrically, it's the point where the graphs of the equations intersect.For the system $y = 2x + 3$ and $y = -x + 6$, the solution is $(1, 5)$ because $5 = 2(1) + 3$ and $5 = -(1) + 6$ are both true.
Graphing a Linear EquationThe process of plottin...
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Core Formulas
Slope-Intercept Form
$y = mx + b$
This form is ideal for graphing. 'm' is the slope (rise over run), and 'b' is the y-intercept (the point where the line crosses the y-axis, (0, b)).
Standard Form of a Linear Equation
$Ax + By = C$
Another common form. To graph from this form, you can find the x- and y-intercepts by setting one variable to zero, or rearrange it into slope-intercept form.
Steps to Solve a System by Graphing
1. Define variables. 2. Write two linear equations from the word problem. 3. Convert equations to slope-intercept form ($y=mx+b$) if necessary. 4. Graph both equations on the same coordinate plane. 5. Identify the coordinates of the intersection point. 6. Check the solution in both original equations. 7. State the solution in the...
4 more steps in this tutorial
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Challenging
Two candles are lit. Candle A's height is modeled by y = -0.5x + 12 and Candle B's by y = -0.25x + 10, where x is hours and y is height in inches. What does the y-intercept of Candle A's equation, (0, 12), represent?
A.The initial height of Candle A was 12 inches.
B.It takes 12 hours for Candle A to burn out.
C.Candle A burns at a rate of 12 inches per hour.
D.The height of Candle A after 12 hours.
Challenging
A student compares a monthly streaming service (y = 15x) with a yearly subscription (y = 100). They graph the system and find the intersection is approximately (6.7, 100). Since you can't pay for 0.7 of a month, how should this solution be interpreted?
A.The yearly plan is cheaper after exactly 6.7 months.
B.The monthly plan becomes more expensive than the yearly plan starting in the 7th month.
C.The yearly plan is always a better deal.
D.The monthly plan is a better deal for the first 6 months only.
Challenging
The cost to print flyers at two stores is shown by lines on a graph. Store A's line passes through (100, $50) and (300, $90). Store B's line passes through (100, $60) and (300, $80). At what number of flyers will the cost be exactly the same?
A.150 flyers
B.250 flyers
C.300 flyers
D.200 flyers
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