8. the equation of a line is $y = -5x + b$.
the line passes through the point $(2, 3)$.
what is the value of $b$?

9. the equation of a line is $y = mx - 4$.
the line passes through the point $(-3, -25)$.
what is the value of $m$?

10. determine the equation of a line that passes through $(4, 4)$ and $(1, -2)$.
write your equation in $y = mx + b$ form.

unit 2: quadratic relations

1. expand.
$2(x + 3)$ | $3x(x - 1)$ | $-4(x - 5)$

2. expand and simplify.
$(x + 1)(x + 2)$ | $(x + 2)(x - 3)$ | $(2x + 2)(x + 1)$
$(2x - 1)(x - 3)$ | $(x + 1)(3x - 2)$ | $(2x - 5)(2x - 5)$

3. expand and simplify.
$2(x + 1)(x + 2)$ | $3(x - 2)(x + 3)$ | $-2(2x - 2)(x - 1)$

Question

8. the equation of a line is $y = -5x + b$.
the line passes through the point $(2, 3)$.
what is the value of $b$?

9. the equation of a line is $y = mx - 4$.
the line passes through the point $(-3, -25)$.
what is the value of $m$?

10. determine the equation of a line that passes through $(4, 4)$ and $(1, -2)$.
write your equation in $y = mx + b$ form.

unit 2: quadratic relations

1. expand.
$2(x + 3)$ | $3x(x - 1)$ | $-4(x - 5)$

2. expand and simplify.
$(x + 1)(x + 2)$ | $(x + 2)(x - 3)$ | $(2x + 2)(x + 1)$
$(2x - 1)(x - 3)$ | $(x + 1)(3x - 2)$ | $(2x - 5)(2x - 5)$

3. expand and simplify.
$2(x + 1)(x + 2)$ | $3(x - 2)(x + 3)$ | $-2(2x - 2)(x - 1)$

Accepted Answer

{ "quality": "clear", "question_count": 15, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Slope-Intercept Form", "Linear Function Evaluation", "Algebraic Problem Solving" ], "new_concepts": [ "Polynomial Expansion", "Distributive Property", "Quadratic Relations" ], "current_concepts": [ "Slope-Intercept Form", "Linear Function Evaluation", "Algebraic Problem Solving", "Polynomial Expansion", "Distributive Property", "Quadratic Relations" ] } ## Solve linear equation problems 8 to 10 $$ \begin{aligned} &\text{Q8: } y = -5x + b \implies 3 = -5(2) + b \implies b = 13\ &\text{Q9: } y = mx - 4 \implies -25 = m(-3) - 4 \implies -3m = -21 \implies m = 7\ &\text{Q10: } m = \frac{-2 - 4}{1 - 4} = \frac{-6}{-3} = 2 \implies y - 4 = 2(x - 4) \implies y = 2x - 4 \end{aligned} $$ ## Expand single-bracket expressions in Unit 2, Q1 $$ \begin{aligned} &2(x + 3) = 2x + 6\ &3x(x - 1) = 3x^2 - 3x\ &-4(x - 5) = -4x + 20 \end{aligned} $$ ## Expand and simplify double-bracket expressions in Unit 2, Q2 $$ \begin{aligned} &(x + 1)(x + 2) = x^2 + 3x + 2\ &(x + 2)(x - 3) = x^2 - x - 6\ &(2x + 2)(x + 1) = 2x^2 + 4x + 2\ &(2x - 1)(x - 3) = 2x^2 - 7x + 3\ &(x + 1)(3x - 2) = 3x^2 + x - 2\ &(2x - 5)(2x - 5) = 4x^2 - 20x + 25 \end{aligned} $$ ## Expand and simplify three-factor expressions in Unit 2, Q3 $$ \begin{aligned} &2(x + 1)(x + 2) = 2(x^2 + 3x + 2) = 2x^2 + 6x + 4\ &3(x - 2)(x + 3) = 3(x^2 + x - 6) = 3x^2 + 3x - 18\ &-2(2x - 2)(x - 1) = -2(2x^2 - 4x + 2) = -4x^2 + 8x - 4 \end{aligned} $$ ## Linear Equations Section | No. | Problem | Answer | |---|---|---| | 8 | Find $b$ for $y = -5x + b$ through $(2, 3)$ | $b = 13$ | | 9 | Find $m$ for $y = mx - 4$ through $(-3, -25)$ | $m = 7$ | | 10 | Find equation through $(4, 4)$ and $(1, -2)$ | $y = 2x - 4$ | ## Unit 2: Quadratic Relations — Question 1 (Expand) | Column 1 | Column 2 | Column 3 | |---|---|---| | $2(x + 3) = 2x + 6$ | $3x(x - 1) = 3x^2 - 3x$ | $-4(x - 5) = -4x + 20$ | ## Unit 2: Quadratic Relations — Question 2 (Expand and simplify) | Column 1 | Column 2 | Column 3 | |---|---|---| | $(x + 1)(x + 2) = x^2 + 3x + 2$ | $(x + 2)(x - 3) = x^2 - x - 6$ | $(2x + 2)(x + 1) = 2x^2 + 4x + 2$ | | $(2x - 1)(x - 3) = 2x^2 - 7x + 3$ | $(x + 1)(3x - 2) = 3x^2 + x - 2$ | $(2x - 5)(2x - 5) = 4x^2 - 20x + 25$ | ## Unit 2: Quadratic Relations — Question 3 (Expand and simplify) | Column 1 | Column 2 | Column 3 | |---|---|---| | $2(x + 1)(x + 2) = 2x^2 + 6x + 4$ | $3(x - 2)(x + 3) = 3x^2 + 3x - 18$ | $-2(2x - 2)(x - 1) = -4x^2 + 8x - 4$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Polynomial Expansion" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Solve linear equation problems 8 to 10

Expand single-bracket expressions in Unit 2, Q1

Expand and simplify double-bracket expressions in Unit 2, Q2

Expand and simplify three-factor expressions in Unit 2, Q3

Answer:

Linear Equations Section

Unit 2: Quadratic Relations — Question 1 (Expand)

Unit 2: Quadratic Relations — Question 2 (Expand and simplify)

Unit 2: Quadratic Relations — Question 3 (Expand and simplify)

No.	Problem	Answer
9	Find \(m\) for \(y = mx - 4\) through \((-3, -25)\)	\(m = 7\)
10	Find equation through \((4, 4)\) and \((1, -2)\)	\(y = 2x - 4\)