Math Problem Statement

Solve the system of equations using the substitution method: x + y = 17 and y = x + 3.

Solution

The given system of equations is:

x + y = 17 \\ y = x + 3 \end{cases}$$ We are asked to solve this system using the substitution method. Let's follow the steps: ### Step 1: Substitute the value of $$y$$ from the second equation into the first equation. From the second equation, we have $$y = x + 3$$. Substituting this into the first equation: $$x + (x + 3) = 17$$ ### Step 2: Solve for $$x$$. Simplify the equation: $$2x + 3 = 17$$ Subtract 3 from both sides: $$2x = 14$$ Now, divide both sides by 2: $$x = 7$$ ### Step 3: Substitute $$x = 7$$ back into the second equation to find $$y$$. Using $$y = x + 3$$: $$y = 7 + 3 = 10$$ ### Solution: The solution to the system is $$(x, y) = (7, 10)$$. Would you like more details on this solution or have any further questions? Here are some related problems you might want to explore: 1. What happens if you use elimination instead of substitution? 2. Solve the system $$x - y = 5$$ and $$y = 2x - 1$$ by substitution. 3. What is the graphical interpretation of the solution to this system? 4. How would the solution change if the system had been $$x + y = 17$$ and $$y = x - 3$$? 5. Can substitution always be used to solve linear systems? Why or why not? **Tip:** Always check your solution by substituting both values into the original equations!

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Substitution Method

Formulas

x + y = 17
y = x + 3

Theorems

Substitution Method

Suitable Grade Level

Grades 6-8