Math Problem Statement

Solve the math problems in the provided worksheet, which includes finding equations of lines, solving a system of equations, and applying the binomial theorem.

Solution

Let's go through each problem in the image step-by-step:


Problem 1: Equations of Lines and Circle

(a) Write the equation of the line passing through points A(5,4)A(5, 4) and B(1,3)B(1, -3).

  1. Calculate the slope mm: m=y2y1x2x1=3415=74=74m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 4}{1 - 5} = \frac{-7}{-4} = \frac{7}{4}
  2. Use point-slope form yy1=m(xx1)y - y_1 = m(x - x_1) with point A(5,4)A(5, 4): y4=74(x5)y - 4 = \frac{7}{4}(x - 5) Expand to find the line equation.

(b) Write the equation of the line passing through (2,3)(-2, -3) and perpendicular to 3x+6y=53x + 6y = 5.

  1. Find the slope of 3x+6y=53x + 6y = 5 by rewriting it in slope-intercept form y=mx+by = mx + b.
  2. The perpendicular slope is the negative reciprocal.
  3. Use point-slope form with point (2,3)(-2, -3) to find the line equation.

(c) Find the area and perimeter of the circle with equation x2+y26x4y=0x^2 + y^2 - 6x - 4y = 0.

  1. Complete the square to rewrite in standard form (xh)2+(yk)2=r2(x - h)^2 + (y - k)^2 = r^2.
  2. Calculate the radius rr, then:
    • Area = πr2\pi r^2
    • Perimeter (circumference) = 2πr2\pi r

Problem 2: Solving Systems and Inequalities

(a) Solve the system of equations:

x = 2 - y^2 \\ 2 = 3x + y \end{cases}$$ Substitute $$x = 2 - y^2$$ into the second equation and solve for $$y$$, then find $$x$$. **(b)** Solve the inequality $$x^2 + x - 6 > 0$$. 1. Factor the quadratic as $$(x + 3)(x - 2) > 0$$. 2. Determine intervals where the inequality holds. --- ### Problem 3: Polynomials and Binomial Theorem **(a)** Given $$(x - 2)$$ as a factor of $$x^3 + x^2 - 2x - 8$$, find the other factor. 1. Perform synthetic or polynomial division to divide $$x^3 + x^2 - 2x - 8$$ by $$x - 2$$. **(b)** Find the term with $$x^7$$ in the expansion of $$(3 + x)^{12}$$. 1. Use the binomial term formula $$\binom{n}{k} a^{n-k} b^k$$. 2. Set up for the term where $$k = 7$$. **(c)** Find the fourth term in $$(3 + x)^{12}$$ using the binomial theorem. 1. Use the formula for the binomial expansion term $$T_{k+1} = \binom{n}{k} a^{n-k} b^k$$ with $$n = 12$$ and $$k = 3$$. --- Would you like detailed solutions for each part, or have specific questions? --- ### Related Questions 1. How do you find the equation of a line through two given points? 2. How do you find a perpendicular line to a given line? 3. What are the steps for completing the square in a circle's equation? 4. How do you solve a system of nonlinear equations? 5. What is the general approach to solving polynomial inequalities? ### Tip For problems involving the binomial theorem, remember the general term formula, $$T_{k+1} = \binom{n}{k} a^{n-k} b^k$$, to quickly find any specific term in the expansion.

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Math Problem Analysis

Mathematical Concepts

Algebra
Coordinate Geometry
Polynomial Division
Systems of Equations
Inequalities
Binomial Theorem

Formulas

Point-slope form of a line: y - y1 = m(x - x1)
Slope formula: m = (y2 - y1) / (x2 - x1)
Circle equation: (x - h)^2 + (y - k)^2 = r^2
Binomial term formula: T(k+1) = C(n, k) * a^(n-k) * b^k

Theorems

Binomial Theorem
Properties of Polynomial Division

Suitable Grade Level

Grades 10-12