a system of linear equations includes the line that is created by the equation $y = x + 3$, graphed below, and the line through the points $(3, 1)$ and $(4, 3)$.

what is the solution to the system of equations?

Question

a system of linear equations includes the line that is created by the equation $y = x + 3$, graphed below, and the line through the points $(3, 1)$ and $(4, 3)$.

what is the solution to the system of equations?

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Slope-Intercept Form", "Line Intersection", "System of Equations" ], "new_concepts": [], "current_concepts": [ "Slope-Intercept Form", "Line Intersection", "System of Equations" ] } ## Find the equation of the second line Using the *Slope-Intercept Form* knowledge point $$ \begin{aligned} m &= \frac{3 - 1}{4 - 3} = 2\ y - 1 &= 2(x - 3)\ y &= 2x - 5 \end{aligned} $$ ## Solve the system of equations Using the *System of Equations* knowledge point $$ \begin{aligned} 2x - 5 &= x + 3\ x &= 8\ y &= 8 + 3 = 11 \end{aligned} $$ ## Verify the intersection point Using the *Line Intersection* knowledge point $$ \begin{aligned} \text{Line 1: } 11 &= 8 + 3 \quad (\text{True})\ \text{Line 2: } 11 &= 2(8) - 5 \quad (\text{True}) \end{aligned} $$ $(8, 11)$ { "elements": [ { "type": "functiongraph", "params": [ { "js": "x + 3", "latex": "y = x + 3" }, -2, 12 ], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2, "name": "y = x + 3", "withLabel": true } }, { "type": "functiongraph", "params": [ { "js": "2*x - 5", "latex": "y = 2x - 5" }, 2, 12 ], "properties": { "strokeColor": "#F2557F", "strokeWidth": 2, "name": "y = 2x - 5", "withLabel": true } }, { "type": "point", "params": [ [3, 1] ], "properties": { "name": "(3, 1)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [4, 3] ], "properties": { "name": "(4, 3)", "size": 3, "color": "#5583F2", "withLabel": true } }, { "type": "point", "params": [ [8, 11] ], "properties": { "name": "(8, 11)", "size": 4, "color": "#55DDF2", "withLabel": true } } ] } { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Algebra", "System of Equations" ] }

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QUESTION IMAGE

Question

Explanation:

Find the equation of the second line

Solve the system of equations

Verify the intersection point

Answer: