Question
How many solutions does the system of equations have? \( \begin{array}{l}y=-\frac{7}{2} x+11 \\ 7 x+2 y=20 \\ \text { A. } 1 \text { solution } \\ \text { B. no solution } \\ \text { C. infinitely many solutions } \\ \text { D. cannot be determined }\end{array} \)
Ask by Elliott Daniel. in the United States
Mar 11,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The system has no solution.
Solution
Solve the system of equations \( 7x+2y=20;y=-\frac{7}{2}x+11 \).
Solve the system of equations by following steps:
- step0: Solve using the substitution method:
\(\left\{ \begin{array}{l}7x+2y=20\\y=-\frac{7}{2}x+11\end{array}\right.\)
- step1: Substitute the value of \(y:\)
\(7x+2\left(-\frac{7}{2}x+11\right)=20\)
- step2: Simplify:
\(22=20\)
- step3: Calculate:
\(\left\{ \begin{array}{l}x \in \varnothing \\y \in \varnothing \end{array}\right.\)
- step4: Rewrite:
\((x, y) \in \varnothing\)
The system of equations has no solution. Therefore, the correct answer is B. no solution.
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Simplify this solution Bonus Knowledge
To determine how many solutions the system of equations has, we can analyze the equations graphically or algebraically. The first equation, \( y = -\frac{7}{2} x + 11 \), represents a line with a slope of \(-\frac{7}{2}\). The second equation can be rewritten in slope-intercept form. Rearranging \( 7x + 2y = 20 \) gives us \( y = -\frac{7}{2}x + 10 \). Now, notice that both lines have the same slope of \(-\frac{7}{2}\) but different y-intercepts (11 and 10, respectively). This means that the lines are parallel and will never intersect, indicating that there is no solution to the system. Therefore, the answer is **B. no solution**.
