Equivalent Linear Relations and Systems
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Questions and Answers

What defines equivalent linear relations in terms of their graphs?

  • They have the same y-intercept and slope.
  • They have infinite points of intersection. (correct)
  • They have no points of intersection.
  • They have different slopes and y-intercepts.

Which scenario represents zero points of intersection for two lines?

  • The slopes are equal and y-intercepts differ. (correct)
  • The slopes are equal and y-intercepts are also equal.
  • The slopes are different but y-intercepts are equal.
  • The slopes are different and y-intercepts differ.

What is the point of intersection (POI) of the lines y = -2x - 5 and y = 3x + 5?

  • (2, 1)
  • (0, -5)
  • (-2, -1) (correct)
  • (1, -4)

What does a perfect square trinomial represent in terms of binomials?

<p>(a + b)^2 (A)</p> Signup and view all the answers

What is the first step in using the FOILING method for the binomial (m + 4)^2?

<p>Square the first variable. (D)</p> Signup and view all the answers

When applying the Greatest Common Factor (GCF) method, what is required?

<p>There must be at least two terms. (D)</p> Signup and view all the answers

How is the polynomial 12x^5 + 3 factored using GCF?

<p>3(4x^5 + 1) (D)</p> Signup and view all the answers

What does a difference of squares most commonly apply to?

<p>A binomial product. (A)</p> Signup and view all the answers

What is the value of n when m is substituted back into the equation 5m + 6n = 15?

<p>$ rac{35}{3}$ (B)</p> Signup and view all the answers

What is the slope of the median line CD in triangle ABC with vertices A(3,4), B(-5,2), and C(1,-4)?

<p>$- rac{7}{2}$ (A)</p> Signup and view all the answers

If each side of a playground measuring 60m by 40m is extended by an equal amount to double the area, what is the equation used to find the amount x of extension?

<p>$(60 + 2x)(40 + 2x) = 4800$ (B)</p> Signup and view all the answers

For the quadratic function $h=0.025d^2 + d$, what method can be used to solve for the maximum height?

<p>Completing the square (A)</p> Signup and view all the answers

What is the relationship of revenue in terms of units sold and price, according to the provided content?

<p>Revenue = (Number of units sold) x (Price per unit). (D)</p> Signup and view all the answers

In the car wash problem, if the total number of cars and trucks is represented by x and y respectively, what does the equation 5x + 10y = 600 represent?

<p>Total money earned based on the vehicle types. (C)</p> Signup and view all the answers

How many trucks went through the car wash if the equations x + y = 100 and 5x + 10y = 600 are solved?

<p>$20$ (C)</p> Signup and view all the answers

What is the standard form of a quadratic function represented in the form y = ax^2 + bx + c?

<p>$y = ax^2 + bx + c$ (C)</p> Signup and view all the answers

Which of the following represents the condition needed for a trinomial to be classified as a perfect square?

<p>The square root of the first term and the last term must equal half the middle term. (B)</p> Signup and view all the answers

What is the result of factoring the trinomial $3m^2 + 10m + 3$ using the decomposition method?

<p>(m + 3)(3m + 1) (B)</p> Signup and view all the answers

What type of polynomial is described by a structure of three terms where the leading coefficient is not equal to 1?

<p>Complex trinomial (C)</p> Signup and view all the answers

How do you factor the expression $x^2 - 16$?

<p>(x + 4)(x - 4) (B)</p> Signup and view all the answers

Which statement accurately describes the range of a parabola that opens downwards?

<p>y ≤ the y-coordinate of the vertex (A)</p> Signup and view all the answers

In the context of the midpoint of a line segment, what does the altitude specifically refer to?

<p>A line that bisects the segment at 90°. (A)</p> Signup and view all the answers

In the equation of a circle given by $x^2 + y^2 = c^2$, what does it indicate if the left side is greater than the right side?

<p>The point is outside of the circle. (A)</p> Signup and view all the answers

What is the proper method to apply when using the substitution method in solving a system of equations?

<p>Isolate one variable and substitute it back into the first equation. (A)</p> Signup and view all the answers

Which law is applied when finding an angle from a triangle given two sides and the included angle?

<p>Cosine Law (C)</p> Signup and view all the answers

What does completing the square allow you to convert?

<p>A complex trinomial into a perfect square trinomial. (D)</p> Signup and view all the answers

If given the polynomial $2(4y^{2}-8y-3y+6)$, which steps should you take to fully factor it?

<p>Factor out the GCF first then look for grouping. (A)</p> Signup and view all the answers

What is the product of the square roots used in the perfect square test?

<p>Square root of the first term multiplied by the square root of the last term. (B)</p> Signup and view all the answers

In the context of quadratic relations, which characteristic describes the axis of symmetry?

<p>The vertex's x-coordinate. (B)</p> Signup and view all the answers

Flashcards

Equivalent Linear Relations

Linear relations that have the same graph and share an infinite number of points of intersection.

Equivalent Linear Systems

Linear systems with the same point of intersection.

Point of Intersection (POI)

The point where two lines intersect.

Solving for the POI

A method to find the POI of two linear equations by setting them equal to each other and solving for the unknown variable.

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Foiling

A mathematical process of multiplying two binomials.

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Perfect Square Trinomial

A special product of a binomial squared, resulting in a trinomial.

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Difference of Squares

A special product of two binomials with opposite signs, resulting in a binomial.

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Factoring by Greatest Common Factor (GCF)

A method to factorize expressions by finding and separating the greatest common factor.

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Midpoint Formula

Finding the midpoint of a line segment using the coordinates of its endpoints.

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Slope Formula

Calculating the steepness of a line using the coordinates of two points on the line.

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Median of a Triangle

A line segment that connects a vertex of a triangle to the midpoint of the opposite side.

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Equation of a Line Formula

Writing an equation for a line based on its slope and a point on the line.

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Elimination Method

Solving a system of equations by adding or subtracting the equations together to eliminate one variable.

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Substitution Method

Solving a system of equations by solving one equation for one variable and substituting it into the other equation.

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Completing the Square

Finding the maximum or minimum value of a quadratic function by rewriting the function in vertex form.

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Revenue Function

The function that describes the revenue earned from selling a product, where revenue is the product of the number of units sold and the price per unit.

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Simple Trinomial

A trinomial whose first term has a coefficient of 1. To factor, find two numbers whose product equals the last term and whose sum equals the middle term.

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Complex Trinomial

A trinomial where the first term has a coefficient other than 1. Factor by decomposition: find two numbers that multiply to the product of the first and last terms, and add up to the middle term. Then, factor by grouping.

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Quadratic Formula

A formula for find the solutions (roots) of a quadratic equation. It states that the solutions are given by: x = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

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Median

A line segment that divides a triangle in half, but does not necessarily intersect at a right angle. It connects a vertex to the midpoint of the opposite side.

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Altitude

A line segment that is perpendicular to a side of a triangle and passes through the opposite vertex. It represents the height of the triangle.

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Perpendicular Bisector

A line segment that bisects a line segment at a 90-degree angle. It cuts a line segment in half and is perpendicular to it.

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Distributive Property

The rule states that when a monomial is outside brackets, you distribute it to each term inside the brackets by multiplication.

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Sine (Sin)

The trigonometric ratio of a right triangle, defined as the ratio of the length of the opposite side to the length of the hypotenuse. It is abbreviated as Sin θ.

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Cosine (Cos)

The trigonometric ratio of a right triangle, defined as the ratio of the length of the adjacent side to the length of the hypotenuse. It is abbreviated as Cos θ.

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Tangent (Tan)

The trigonometric ratio of a right triangle, defined as the ratio of the length of the opposite side to the length of the adjacent side. It is abbreviated as Tan θ.

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Sine Law

A rule that relates the sides and angles of a triangle. It states that the ratio of a side to the sine of its opposite angle is constant for all three sides of the triangle.

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Cosine Law

A rule that relates the sides and angles of a triangle. It states that the square of any side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them.

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Study Notes

Equivalent Linear Relations and Systems

  • Equivalent linear relations have infinitely many points of intersection.
  • Equivalent linear relations share the same graph.
  • Creating infinitely many equivalent linear relations: Multiply or divide the entire equation by the same non-zero number.
  • One point of intersection (POI): Different slopes, different y-intercepts. (e.g., y = 5x + 7, y = 3x - 2)
  • Zero points of intersection: Same slope, different y-intercepts. (e.g., y = 4x + 10, y = 4x + 3)
  • Infinite points of intersection: Lines coincide. Same slopes and same y-intercepts. (e.g., y = 3x + 5, 2y = 6x + 10)
  • Equivalent linear systems share the same point of intersection.

Finding the Point of Intersection (POI)

  • Convert both equations to slope-intercept form (y = mx + b).
  • Set the expressions for 'y' equal to each other and solve for 'x'.
  • Substitute the 'x' value back into either original equation to find the 'y' value.

Example: Finding the POI

  • Given: y = -2x - 5 and y = 3x + 5
  • Set them equal: -2x - 5 = 3x + 5
  • Solve for x: x = -2
  • Substitute x = -2 into either equation (e.g., y = 3x + 5): y = 3(-2) + 5 = -1
  • POI: (-2, -1)

Foiling

  • Foiling is multiplying two binomials.
  • Example: (x + 4)(x + 5) = x² + 9x + 20
  • Perfect Square Trinomial: (a + b)² = a² + 2ab + b²
  • Difference of Squares: (a - b)(a + b) = a² - b²

Foiling (Powers)

  • For squaring binomials, a shortcut exists:
  • Square the first term, double the product of the terms, and square the last term.
  • Example: (m + 4)² = m² + 8m + 16

Factoring

  • Factoring reverses foiling.
  • Methods: Greatest Common Factor (GCF), Common Brackets, Grouping, Trinomials (Perfect Square, Simple, Complex), and Difference of Squares.

Greatest Common Factor (GCF)

  • Find the GCF of all terms.
  • Divide all terms by the GCF.
  • Example: 12x⁵ + 3 = 3(4x⁵ + 1) , and 14m⁶ - 7m⁴ + 21m² = 7m²(2m⁴ - m² + 3)

Grouping

  • Use if no overall GCF exists. Group terms with common factors, then find GCF of the groups.
  • Example: m² - 4n + 4m - mn = (m + 4)(m - n), and xy + 12 + 4x + 3y = (x + 3)(y + 4)

Perfect Square Trinomial Factoring

  • For expressions that follow the form a² + 2ab + b²
  • Square root first and last term to find the binomial
  • Example: 16t² + 24t + 9 = (4t+3)² , and 3x² + 6x + 3 = 3(x+1)²

Simple Trinomial Factoring

  • If the first term's coefficient is 1, find factors of the last term whose sum equals the middle term.
  • Example: x² + 8x + 15 = (x + 5)(x + 3) and p² - 8p - 20 = (p - 10)(p + 2)

Complex Trinomial Factoring

  • If the first term's coefficient isn't 1, use the decomposition method to find two numbers whose product equals ac and sum equals b.
  • Example: 3m² + 10m + 3 = (m + 3)(3m + 1) and 8y² - 22y + 12 = 2(y - 2)(4y - 3)

Difference of Squares Factoring

  • Must have two terms with a subtraction.
  • Example: x² - 16 = (x+4)(x-4) and (x-4)²-(x+3)²=(2x-1)(-7)

Equation of a Circle

  • Origin centered: x² + y² = r²
  • To determine if a point is inside, outside, or on the circle: Substitute coordinates into the equation.

Midpoint of a Line Segment

  • The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂): ((x₁ + x₂)/2, (y₁ + y₂)/2).

Exponent Laws

  • Multiplication: Add exponents. xm × xn = xm+n
  • Division: Subtract exponents. xm ÷ xn = xm-n
  • Power of a power: Multiply exponents. (xm)n = xmn
  • Fractional exponents: (1/m)n = 1n/mn
  • Zero exponent: x⁰ =1
  • Negative exponent: x⁻ⁿ = 1/xⁿ

Trigonometric Ratios

  • Right-angled triangles: SOH CAH TOA
  • Sin θ = opposite/hypotenuse
  • Cos θ = adjacent/hypotenuse
  • Tan θ = opposite/adjacent

Sine Law

  • Find sides or angles in triangles with known opposite sides and angles. a / Sin A = b / Sin B = c / Sin C

Cosine Law

  • Find sides or angles in triangles with known sides and angles. a² = b² + c² - 2bc Cos A b² = a² + c² - 2ac Cos B c² = a² + b² - 2ab Cos C

Completing the Square (Quadratic Equations)

  • Convert from standard form to vertex form.

Descriptive Table (Quadratic Relations)

  • Table of information about a quadratic relationship: vertex, direction of opening, axis of symmetry, minimum/maximum value, domain, and range.

Quadratic Formula

  • Solving quadratic equations.

Distributive Property

  • Multiplying a monomial by a binomial; expand the expression by multiplying all terms. 5(x + 3) = 5x + 15

Substitution Method (Systems of Equations)

  • Solve one equation for a single variable.
  • Substitute the solution into the second equation.
  • Solve for the remaining variable.
  • Substitute back to find the other variable.

Elimination Method (Systems of Equations)

  • Express equations in ax + by = c form.
  • If necessary, create equivalent equations to have opposite coefficients for one variable.
  • Add or subtract the equations to eliminate one variable.

Length of a Line Formula

  • Calculate the distance between points in a coordinate plane

Extending/Shrinking Sides

  • Increase the area of figures proportionally by increasing each side equally then use algebra to solve.

Ball Problem

  • Use the quadratic equation provided to determine the maximum height or when a specific height is reached.

Revenue

  • Calculate revenue (price per unit times units sold).

Car Problem

  • Use a system of equations to find the number of cars and trucks.

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Description

Explore the concepts of equivalent linear relations and systems in this quiz. Learn how to determine points of intersection and understand the conditions for equivalent lines. This material is essential for mastering linear equations and their graphical representations.

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