Math Problem Statement
The sum of two numbers is 2525 and their difference is −5−5. What are the two numbers? Let x=x= one number and y=y= the other number. The corresponding modeling system is {x+y=25x−y=−5{x+y=25x−y=−5. Solve the system graphically.
Solution
We are given a system of equations representing two numbers and :
1) & \quad x + y = 25 \\ 2) & \quad x - y = -5 \end{aligned}$$ ### Step 1: Solve the system algebraically We can solve this system of equations by adding or subtracting them. #### Adding the two equations: $$(x + y) + (x - y) = 25 + (-5)$$ This simplifies to: $$2x = 20$$ Solving for $$x$$: $$x = 10$$ #### Substituting $$x = 10$$ into the first equation: $$10 + y = 25$$ Solving for $$y$$: $$y = 25 - 10 = 15$$ So the two numbers are $$x = 10$$ and $$y = 15$$. ### Step 2: Graphically solving the system 1. **Graph of the first equation**: $$x + y = 25$$ - Rearrange to: $$y = 25 - x$$ - This is a line with a slope of -1 and a y-intercept of 25. 2. **Graph of the second equation**: $$x - y = -5$$ - Rearrange to: $$y = x + 5$$ - This is a line with a slope of 1 and a y-intercept of 5. When these two lines are plotted, their point of intersection gives us the solution to the system: $$x = 10$$ and $$y = 15$$. Would you like me to plot the graph for you, or do you have any other questions about this? ### Further Questions: 1. How would you solve a system of equations with three variables? 2. What is the geometric interpretation of the system of equations involving two lines? 3. How do changes in the coefficients of a system affect the graphs of the lines? 4. Can you solve this system using substitution instead of elimination? 5. How do you interpret the slope of a line in the context of this system? ### Tip: To solve systems graphically, always rearrange equations to the slope-intercept form ($$y = mx + b$$) so you can easily plot them on a graph.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Graphical Methods
Formulas
x + y = 25
x - y = -5
Theorems
Solving Systems of Linear Equations
Suitable Grade Level
Grades 8-10