Math Problem Statement
Graph the system of equations: y = (1/2)x + 1 and x - 2y = -2, and write its solution.
Solution
We are given the following system of equations:
y &= \frac{1}{2}x + 1 \\ x - 2y &= -2 \end{aligned}$$ ### Step 1: Solve the system algebraically To solve the system, we can use substitution or elimination. I'll start by using **substitution** since the first equation is already solved for $$y$$. 1. From the first equation: $$y = \frac{1}{2}x + 1$$ 2. Substitute this expression for $$y$$ into the second equation: $$x - 2\left(\frac{1}{2}x + 1\right) = -2$$ ### Step 2: Simplify and solve for $$x$$ Expand and simplify the equation: $$x - 2\left(\frac{1}{2}x + 1\right) = -2$$ $$x - x - 2 = -2$$ $$-2 = -2$$ This simplifies to a **true statement** $$-2 = -2$$, meaning that the two lines are actually the same line. Therefore, this system has **infinitely many solutions**. ### Step 3: Graph the system Since both equations represent the same line, graphing them would show a single line, and every point on this line is a solution to the system. ### Solution The system has **infinitely many solutions** because the two equations describe the same line. --- Would you like more details or have any questions? Here are five related questions: 1. How do you graph linear equations on a coordinate plane? 2. What does it mean when two lines are the same on a graph? 3. How do you check for parallel or coincident lines in a system of equations? 4. What is the difference between substitution and elimination methods? 5. How can you determine if a system of equations has no solution? **Tip**: Whenever you get a true statement like $$0 = 0$$ or $$-2 = -2$$, it indicates that the system has infinitely many solutions!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Equations
System of Equations
Algebra
Formulas
Slope-Intercept Form: y = mx + b
Substitution Method
Theorems
Coincident Lines in a System of Equations
Suitable Grade Level
Grades 8-10