34. which linear system has an infinite number of solutions?

a) $\begin{cases} y = -2x + 4 \\ 4x + 2y = 2 \end{cases}$

b) $\begin{cases} y = -2x + 4 \\ 4x + 2y = 8 \end{cases}$

c) $\begin{cases} y = 2x + 4 \\ 4x + 2y = 2 \end{cases}$

d) $\begin{cases} y = 2x + 4 \\ 4x + 2y = 8 \end{cases}$

Question

34. which linear system has an infinite number of solutions?

a) $\begin{cases} y = -2x + 4 \\ 4x + 2y = 2 \end{cases}$

b) $\begin{cases} y = -2x + 4 \\ 4x + 2y = 8 \end{cases}$

c) $\begin{cases} y = 2x + 4 \\ 4x + 2y = 2 \end{cases}$

d) $\begin{cases} y = 2x + 4 \\ 4x + 2y = 8 \end{cases}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Slope-Intercept Form", "Linear Equations" ], "new_concepts": [ "Systems of Linear Equations", "Infinite Solutions" ], "current_concepts": [ "Slope-Intercept Form", "Linear Equations", "Systems of Linear Equations", "Infinite Solutions" ] } ## Define conditions for infinite solutions Using the *Linear Equations* knowledge point $$ \begin{aligned} &\text{Two equations represent the same line.}\ &\text{Their slopes and y-intercepts must be equal.} \end{aligned} $$ ## Convert first equation to slope-intercept form Using the *Slope-Intercept Form* knowledge point $$ \begin{aligned} &y = mx + b\ &\text{Equation 1 in all options is already in this form.} \end{aligned} $$ ## Rewrite second equation in slope-intercept form Using the *Linear Equations* and *Slope-Intercept Form* knowledge points $$ \begin{aligned} &4x + 2y = C\ &2y = -4x + C\ &y = -2x + \frac{C}{2} \end{aligned} $$ ## Compare equations for each option Using the *Linear Equations* knowledge point $$ \begin{aligned} &\text{For Option A: } y = -2x + 4 \text{ and } y = -2x + 1 \quad (\text{Parallel, no solution})\ &\text{For Option B: } y = -2x + 4 \text{ and } y = -2x + 4 \quad (\text{Identical, infinite solutions})\ &\text{For Option C: } y = 2x + 4 \text{ and } y = -2x + 1 \quad (\text{Intersecting, one solution})\ &\text{For Option D: } y = 2x + 4 \text{ and } y = -2x + 4 \quad (\text{Intersecting, one solution}) \end{aligned} $$

(A) $\begin{cases} y = -2x + 4 \ 4x + 2y = 2 \end{cases}$

(B) $\begin{cases} y = -2x + 4 \ 4x + 2y = 8 \end{cases}$

(C) $\begin{cases} y = 2x + 4 \ 4x + 2y = 2 \end{cases}$

(D) $\begin{cases} y = 2x + 4 \ 4x + 2y = 8 \end{cases}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Systems of Linear Equations" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Define conditions for infinite solutions

Convert first equation to slope-intercept form

Rewrite second equation in slope-intercept form

Compare equations for each option

Answer: