Algebra - How To Solve Equations Quickly!

The Organic Chemistry Tutor
15 Jul 201725:04

TLDRThis video tutorial offers a comprehensive guide to solving two-step algebraic equations swiftly. It covers equations with fractions, parentheses, variables on both sides, and even those involving decimals. The instructor demonstrates solving equations by isolating variables, using opposite operations like subtraction and division, and combining like terms. Examples include dealing with parentheses, moving variables to one side, and eliminating decimals by scaling. The video is an excellent resource for mastering basic algebraic equation solving techniques.

Takeaways

  • 📘 Solving two-step equations involves isolating the variable, often by performing operations such as subtraction or division on both sides of the equation.
  • 🔢 When dealing with equations that have constants added to variables, subtract the constant from both sides to isolate the variable term.
  • ➗ To separate the variable from a coefficient, use the inverse operation; if the variable is multiplied by a number, divide both sides by that number.
  • 🔄 When an equation contains a variable on both sides, move all variable terms to one side and constants to the other to simplify the equation.
  • 📉 For equations with parentheses, use the distributive property to eliminate them, multiplying the term outside the parentheses to each term inside.
  • 📖 When encountering fractions, multiply every term by the least common multiple (LCM) of the denominators to clear the fractions.
  • 🔄 Similar to fractions, decimals can be eliminated by multiplying all terms by a power of 10 to shift the decimal places, effectively converting them to whole numbers.
  • 📌 To check your work, substitute the solved variable value back into the original equation to ensure both sides balance.
  • 🔢 In more complex equations with multiple variables or terms, combine like terms before isolating the variable to simplify the process.
  • 📘 The video also promotes an algebra course on Udemy, suggesting further resources for those seeking to deepen their understanding of algebraic concepts.

Q & A

  • What is the first step in solving a two-step equation like 3x + 5 = 17?

    -The first step in solving a two-step equation like 3x + 5 = 17 is to isolate the variable x. This is done by performing the opposite operation of what is done to x. Since 5 is added to 3x, you would subtract 5 from both sides of the equation.

  • How do you solve the equation 4x + 3 = 19?

    -To solve 4x + 3 = 19, you start by subtracting 3 from both sides to get 4x = 16. Then, you divide both sides by 4 to isolate x, resulting in x = 4.

  • What is the process for solving an equation with a variable on both sides, such as 17 - 5x = 2?

    -To solve an equation with a variable on both sides, like 17 - 5x = 2, you first eliminate the constant on the side with the variable by performing the opposite operation. Here, you subtract 17 from both sides. Then, you divide by the coefficient of the variable to isolate it, which gives x = 3.

  • How do you handle an equation with a fraction, for example, 9 = 3 + x/4?

    -For an equation with a fraction like 9 = 3 + x/4, you first eliminate the constant on the same side as the fraction by subtracting it from both sides. Then, you multiply both sides by the denominator of the fraction to eliminate it, which in this case is 4, resulting in x = 24.

  • What is the strategy for solving equations with parentheses, such as 3(2x - 4) + 1 = 7?

    -To solve an equation with parentheses like 3(2x - 4) + 1 = 7, you use the distributive property to eliminate the parentheses. You multiply the term outside the parentheses by each term inside, then combine like terms and solve the resulting equation.

  • How do you approach an equation with multiple x terms on one side, for instance, 3x + 8 + 5x = 32?

    -When you have an equation with multiple x terms on one side, such as 3x + 8 + 5x = 32, you first combine like terms by adding the x terms together to get a single x term. Then you proceed with solving the equation as you would with a simpler form.

  • What if you have an equation with variables on both sides and constants on both sides, like 5x + 8 = 2x + 10?

    -For an equation with variables and constants on both sides, you aim to get all the variable terms on one side and all the constant terms on the other. You do this by performing the same operation on both sides to cancel out terms, and then solve for the variable.

  • How do you solve equations with decimals involved, such as 2x + 0.3 = 1.5?

    -To solve equations with decimals, you can either solve them directly or eliminate the decimals by multiplying every term by a power of 10 to make the decimals whole numbers. For 2x + 0.3 = 1.5, multiplying through by 10 gives 20x + 3 = 15, and then you solve for x.

  • What is the approach to solving equations with multiple fractions, like 2x/3 + 4/5 = 3x/2 + 1?

    -When dealing with multiple fractions, you find a common denominator to combine them into a single fraction, or you can eliminate the fractions by multiplying every term by the least common multiple of the denominators. Then, you solve the resulting equation.

  • Can you provide a tip for checking your work after solving an equation?

    -A good tip for checking your work after solving an equation is to substitute your found solution back into the original equation. If the equation balances true, then your solution is likely correct.

Outlines

00:00

📘 Introduction to Solving Two-Step Equations

This segment introduces the topic of solving two-step equations, which encompass equations with fractions, parentheses, variables on both sides, and decimals. The presenter begins with the fundamental two-step equations, demonstrating how to isolate the variable 'x' by performing operations such as subtraction and division. An example equation, 3x + 5 = 17, is used to illustrate the process of isolating 'x' by first subtracting 5 from both sides and then dividing by 3 to solve for 'x'. The segment emphasizes the importance of performing opposite operations to isolate the variable and provides another example for the viewers to practice.

05:02

🔢 Advanced Two-Step Equation Techniques

The second paragraph delves into more complex two-step equations, including those with variables on both sides of the equation. The presenter outlines strategies for dealing with equations that require moving all variable terms to one side and constants to the other, using examples such as 17 - 5x = 2 and 9 = 3 + x/4. The process involves subtracting constants and multiplying both sides by the reciprocal of the fraction to isolate the variable. The segment also covers checking solutions by substituting back into the original equation, ensuring the equation balances, which validates the solution.

10:04

📚 Multi-Step Equations and Combining Like Terms

This part of the script addresses multi-step equations that may involve combining like terms before performing the two-step process. The presenter guides through an example with multiple 'x' terms, 3x + 8 + 5x = 32, emphasizing the need to combine like terms first. The process involves subtracting constants and dividing by the coefficient of 'x' to solve for the variable. The segment reinforces the concept with another similar problem, highlighting the adaptability of the techniques to different forms of equations.

15:10

🔄 Solving Equations with Variables on Both Sides

The fourth paragraph tackles equations where the variable 'x' appears on both sides, necessitating a strategic approach to isolate all 'x' terms on one side. Using an example like 13 - 2x = 4x, the presenter shows how to add 'x' terms to both sides and add constants to the other side to balance the equation. This method clears one side of the variable, allowing for straightforward division to solve for 'x'. The segment also encourages viewers to try a similar problem to practice this technique.

20:13

📖 Handling Equations with Parentheses and Distributive Property

The final paragraph in the script introduces equations that include parentheses, requiring the use of the distributive property to simplify before solving. Examples like 3 * (2x - 4) + 1 = 7 are used to demonstrate how to distribute the multiplier across the terms inside the parentheses and then combine like terms. The presenter then guides through the process of adding or subtracting to isolate the variable, followed by division to find the value of 'x'. The segment also briefly mentions an algebra course available on Udemy, suggesting it as a resource for further learning.

Mindmap

Keywords

💡Two-step equations

Two-step equations are algebraic equations that require two operations to solve for the variable. In the context of the video, these equations are foundational and involve simple manipulations such as addition, subtraction, multiplication, or division. For example, the equation '3x + 5 = 17' is a two-step equation where the first step might involve subtracting 5 from both sides, and the second step would be dividing by 3 to isolate x.

💡Isolate the variable

Isolating the variable is a fundamental step in solving algebraic equations. It involves manipulating the equation to get the variable alone on one side of the equals sign. The video emphasizes this by showing how to move constants or coefficients to the other side of the equation to isolate 'x', as seen when solving '3x + 5 = 17' by first subtracting 5 and then dividing by 3.

💡Opposite operations

In algebra, performing the opposite operation is crucial for solving equations. If an equation has addition or multiplication with the variable, the opposite (subtraction or division, respectively) is used to cancel out the term and isolate the variable. The video script illustrates this by subtracting 5 from both sides of '3x + 5 = 17' to cancel the addition.

💡Like terms

Like terms in algebra are terms that have the same variable raised to the same power. Combining like terms is a common step in solving algebraic equations, as it simplifies the equation. The video mentions this when it instructs to combine '3x' and '5x' into '8x', showing how adding like terms simplifies the equation to a single variable term.

💡Distributive property

The distributive property is a fundamental algebraic principle that allows for the multiplication of a term by each term within a parenthesis. The video script refers to this property when solving equations with parentheses, such as '3 * (2x - 4)', where the '3' is distributed to both '2x' and '-4'.

💡Fractions in equations

Equations can include fractions, which require special handling to solve. The video discusses eliminating fractions by finding a common denominator or by multiplying through by a common multiple to clear the fractions, as shown in the equation '2x/3 + 4/2 = 3x/2 + 1', where a common multiple of 6 is used to eliminate the fractions.

💡Decimals in equations

Algebraic equations may also involve decimals, which can be removed by multiplying all terms by a power of 10 to shift the decimal places. The video provides an example of this with the equation '2x + 0.3 = 1.5', where everything is multiplied by 10 to convert the equation to '20x + 3 = 15', making it easier to solve.

💡Variables on both sides

When variables appear on both sides of an equation, a common strategy is to move all variable terms to one side and all constant terms to the other. The video demonstrates this by moving '5x' from the left to the right side and adding '8' to both sides to isolate the variable on one side, as in the equation '5x + 8 = 4 + 8x'.

💡Least common multiple (LCM)

The least common multiple is the smallest number that is a multiple of two or more numbers. In the context of equations with fractions, finding the LCM of the denominators allows for the elimination of fractions by multiplying through by the LCM. The video explains using the LCM when dealing with equations like '3x/4 + 2/5 = 8/20', where the LCM of 4 and 5 is 20.

💡Order of operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which calculations should be performed. The video touches on this when solving equations that involve multiple operations, ensuring that multiplication and division are performed before addition and subtraction.

Highlights

Focus on solving two-step equations using various algebraic techniques.

Learn to handle equations with fractions, parentheses, and variables on both sides.

Isolate the variable 'X' by performing opposite operations to both sides of the equation.

Use subtraction to eliminate constants and simplify the equation.

Divide by the coefficient of 'X' to solve for the variable.

Practice solving equations with a variable on both sides by moving all variables to one side.

Combine like terms to simplify equations with multiple instances of the variable.

Eliminate fractions by finding a common denominator or multiplying through by a common multiple.

Use the distributive property to simplify equations with parentheses.

Check your work by substituting the solution back into the original equation.

Deal with decimals by multiplying through by a power of 10 to eliminate them.

Solve equations with multiple fractions by finding a least common multiple of the denominators.

Handle equations with variables in both the numerator and denominator by isolating terms.

Explore algebra courses on platforms like Udemy for further learning.

Understand the order of operations and how it applies to solving equations.

Learn to graph linear equations and understand concepts like slope and intercept.

Master solving inequalities and absolute value expressions.

Explore the world of polynomials, factoring, and systems of equations.

Dive into quadratic equations, their graphs, and solving techniques.

Understand complex numbers, exponential functions, and logarithms.

Learn about functions, conic sections, and sequences in algebra.

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