Determine the solution to the system of equations.
Understand the Problem
The question asks to determine the solution to a system of equations represented by the graphs of two hangers, identified as Hanger A and Hanger B. The equations are y = 3x for Hanger A and y = x + 3 for Hanger B. This involves finding the point where the two lines intersect.
Answer
The solution to the system of equations is \( \left( \frac{3}{2}, \frac{9}{2} \right) \).
Answer for screen readers
The solution to the system of equations is ( \left( \frac{3}{2}, \frac{9}{2} \right) ).
Steps to Solve
- Set the equations equal to each other
Since both equations represent $y$, we can set them equal: $$ 3x = x + 3 $$
- Rearrange the equation
To isolate $x$, subtract $x$ from both sides: $$ 3x - x = 3 $$
This simplifies to: $$ 2x = 3 $$
- Solve for $x$
Now, divide both sides by 2: $$ x = \frac{3}{2} $$
- Substitute $x$ back into one of the original equations
Choose either equation to find $y$. We will use Hanger A's equation: $$ y = 3x = 3 \left( \frac{3}{2} \right) = \frac{9}{2} $$
- Final solution
The solution to the system of equations is the point of intersection: $$ \left( \frac{3}{2}, \frac{9}{2} \right) $$
The solution to the system of equations is ( \left( \frac{3}{2}, \frac{9}{2} \right) ).
More Information
The intersection point of the two lines indicates the values of (x) and (y) for which both equations are simultaneously true. This point can represent a balance or equilibrium point in real-world scenarios modeled by these equations.
Tips
- Forgetting to set equations equal: Always remember to equate the expressions for (y) when solving systems.
- Incorrectly simplifying fractions: Ensure all algebraic steps involve careful simplification to avoid errors.
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