Solving Linear Equations

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Questions and Answers

Explain in your own words, why equations with variables on both sides can be useful in comparing costs of real-world situations.

Equations with variables on both sides allow us to model different cost structures (e.g., different initial fees and rates) and find when the total costs are equivalent.

In the equation $5x + 3 = 2x + 9$, what is the first step you would take to isolate the variable terms on one side?

Subtract 2x from both sides of the equation.

In the context of solving equations with variables on both sides, explain the significance of performing the same operation on both sides of the equation.

Maintaining equality; ensures the equation remains balanced and the solution set does not change.

How does the concept of inverse operations help in solving equations with variables on both sides?

<p>Inverse operations are used to isolate the variable by 'undoing' the operations applied to it.</p> Signup and view all the answers

What is a rational number? Give an example.

<p>A rational number is a number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. Example: 1/2.</p> Signup and view all the answers

Explain why zero is considered a rational number.

<p>Zero can be written as a fraction with a non-zero denominator (e.g., 0/1), fitting the definition of a rational number.</p> Signup and view all the answers

What is the reciprocal of $\frac{3}{5}$, and how is it used when solving equations involving fractions?

<p>The reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$. It's used to isolate a variable multiplied by the fraction by multiplying both sides of the equation by the reciprocal.</p> Signup and view all the answers

State the addition property of equality and provide a simple example.

<p>If a = b, then a + c = b + c. Adding the same number to both sides of an equation maintains equality. Example: If x = 5, then x + 2 = 5 + 2.</p> Signup and view all the answers

Explain how the multiplication property of equality is applied when solving an equation like $\frac{x}{4} = 7$.

<p>Multiply both sides of the equation by 4 to isolate x: $\frac{x}{4} * 4 = 7 * 4$, which simplifies to $x = 28$.</p> Signup and view all the answers

When solving an equation with rational numbers, what is the purpose of finding the least common multiple (LCM) of the denominators?

<p>The LCM is used to eliminate the fractions by multiplying both sides of the equation by the LCM, which simplifies the equation to use whole numbers.</p> Signup and view all the answers

What is the first step you should take to solve the equation $2(x + 3) = 5x - 6$?

<p>Apply the distributive property to expand the left side of the equation, resulting in $2x + 6 = 5x - 6$.</p> Signup and view all the answers

Solve the following equation: $4(x + 2) - 2 = 2x + 8$.

<p>$x = 1$</p> Signup and view all the answers

Describe the difference between an equation with 'no solution' and one with 'infinitely many solutions'.

<p>An equation with no solution results in a contradiction (e.g., 2 = 3), indicating no value of the variable makes the equation true. Infinitely many solutions results in an identity (e.g., 0 = 0), meaning any value of the variable makes the equation true.</p> Signup and view all the answers

Explain why the equation $x + 5 = x + 5$ has infinitely many solutions.

<p>Because no matter what value is substituted for <em>x</em>, the equation will always be true.</p> Signup and view all the answers

Explain why the equation $x + 2 = x + 3$ has no solution.

<p>Because for any x, x + 2 will never equal x + 3, or to put it another way, because if you subtract <em>x</em> from both sides, you get 2 = 3, which is false.</p> Signup and view all the answers

Given the equation $3(x - 2) = 3x - 6$, how many solutions does it have? Explain why.

<p>Infinitely many solutions because, after applying the distributive property, the equation simplifies to $3x - 6 = 3x - 6$, which is always true.</p> Signup and view all the answers

If solving a linear equation leads to the statement $0 = 5$, what does this indicate about the solution to the equation?

<p>The equation has no solution.</p> Signup and view all the answers

Solve for x: $4x - 6 = 2x + 2$

<p>$x = 4$</p> Signup and view all the answers

Solve for x: $x + 4 = 19 - 2x$

<p>$x = 5$</p> Signup and view all the answers

Flashcards

Solving Equations with Variables on Both Sides

Isolate variable terms on one side and constants on the other side of the equation.

Rational Number

Real number in the form of p/q, where q is not equal to zero.

Multiplicative Inverse

The number you multiply a term by to get 1.

Addition Property of Equality

Adding equal values to both sides does not change equality.

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Multiplication Property of Equality

Multiplying both sides of an equation by the same number.

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Distributive Property

Distribute the number outside paratheses to each term insider.

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No Solution

Equation has no solution.

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Infinitely Many Solutions

A one-variable linear equation may have infinite solutions.

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Study Notes

  • Welcome to Math Class

Workplan

  • The class routine
  • Get the paper
  • Work on the math puzzle of the day
  • Discuss solving linear equations
  • Solve me activity
  • Wrap up

ARMY CHECK!

  • A: Align yourself on the side in 5 counts
  • R: Roam around your eyes and pick up trash
  • M: Make sure that the alignment of your chair is in place
  • Y: You must smile and greet your teacher and classmates

Routine

  • Prayer
  • Army Check
  • Attendance
  • IXL Skills
  • Task Progress

Recap on Functions

  • Linear functions have the general form y = mx + b
  • Quadratic functions have the general form y = ax² + bx + c
  • Cubic functions have the general form y = ax³ + bx² + cx + d

Equations with Variables on Both Sides

  • Equations with variables on both sides are useful for comparing costs in real-world scenarios.
  • Use inverse operations to solve these equations, isolating variable terms on one side

Solving equations

  • Isolate the terms containing variables on one side and constants on the other.
  • Variable terms may be grouped on either side of the equation.

Rent-a-Car scenario

  • Andy's Rental Car: an initial fee of $20 plus an additional $30 per day to rent a car.
  • Buddy's Rental Car: an initial fee of $36 plus an additional $28 per day.
  • To determine the # of days for which the total cost charged by the companies would be the same:
    • Write an equation representing the total cost of renting a car from Andy's Rental Car: initial Fee + Cost for x days (20 + 30x)
    • Write an equation representing the total cost of renting a car from Buddy's Rental Car: initial Fee + Cost for x days (36 + 28x)
    • Write an equation that can be solved to find the number of days for which the total cost charged by the companies would be the same: Total cost at Andy's + Total cost at Buddy's (20+30x = 36 + 28x)
    • Solving for x for equation 20+30x = 36 + 28x results in the total cost is the same if the rental is for 8 days

Exercises

  • 4x-6=2x+2
  • 3x - 2 = x + 10
  • x + 4 = 19 - 2x

Equations with Rational Numbers

  • A Rational number is a type of real number, which is in the form of p/q where q is not equal to zero.
  • Any fraction with non-zero denominators is a rational number.
    • Examples of rational numbers are 1/2, 1/5, 3/4, and so on.
  • The number "0" is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc.
  • 1/0, 2/0, 3/0, etc. are not rational, since they give infinite values.

Key Concepts

  • Reciprocal (Multiplicative Inverse)
  • Addition Property of Equality: [If a = b, then a + c = b + c ]
  • Multiplication Property of Equality: [If a = b, then a.c = b.c]
  • Computation: When dividing by a fraction - multiply by the reciprocal

Solving Equations

  • An example rational equation is: (7/10)n + (3/2)=(3/5)n + 2
    • Determine the least common multiple of the denominators: 10
    • Multiply both sides of the rational equation by the LCM (10)
    • The product and working results in 7n + 15 = 6n + 20
    • Use inverse operations to solve the equation.
    • A final answer is achieved when n = 5 by subtracting 15 from both sides, then subtracting 6n from both sides.

Equations with Distributive Property

  • 3(x-5)+1=2+x
    • Use the Distributive Property
    • Distribute 3 to the terms inside the parentheses.
    • Simplify.
    • Use inverse operations to solve the equation.
    • Subtract x from both sides
    • Add 14 to both sides.
    • Divide both sides by 2.
  • 5-7k=-4(k+1)-3
    • Distribute -4 to the terms inside the parentheses.
    • Simplify
    • Add 4k to both sides.
    • Subtract 5 from both sides.
    • Divide both sides by -3
    • 9 - (9x - 6) = 3 (4x + 6)

Let's try this and exercises

  • x - 5 = 3
  • y + 7 = 12
  • a - 10 = 4
  • 4(x + 8) - 4 = 34 - 2x

IXL QUIZ

Linear Equations with Many/No Solutions

  • A one-variable linear equation may have no solution, one solution, or infinitely many solutions
    • The equation x+4=x+2 has no solution because it is false for every real number.
    • The equation x+3=3+x has infinitely many solutions because it is true for every real number.

Types of solutions for expressions

  • One Solution x = a where value of x is a, the equation is a true expression
  • Many Solutions a = a where any value of x makes makes the equation a true expression
  • No Solution a ≠ b where there is no value of x to make a true statement

Examples

  • One solution: -7 + 3 = 2x + 2 results in x= 9 for a single solution
  • Many solutions to 7x + 2 = 2x + 2 – 9x , results in 0=0
  • No Solutions -7 + 3 = 2x + 2 - 9x, results in 3=2
    • 3x-12=3(x-4)
    • 3-5n=-5n+121

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