Algebra 1: Solving Basic Equations Flashcards

Questions and Answers

How do you solve an equation like x - 5 = 2?

Add 5 to both sides to isolate x, resulting in x = 7.

How do you solve an equation like x + 6 = 10?

Subtract 6 from both sides to isolate x, resulting in x = 4.

How do you solve an equation like 5x = 35?

Divide both sides by 5 to isolate x, resulting in x = 7.

How do you solve an equation like x/4 = 8?

Multiply both sides by 4 to isolate x, resulting in x = 32. Signup and view all the answers

How do you solve an equation like 3/5x = 2?

Multiply both sides by 5/3 to isolate x, resulting in x = 10/3. Signup and view all the answers

If x = 1/2z and 3y = 5z, which of the following relationships holds between x and y for each nonzero value of z?

10x = 3y Signup and view all the answers

How do you solve x² = 16?

Take the square root of both sides, resulting in x = ±4. Signup and view all the answers

How do you solve x³ = 8?

Take the cube root of both sides, resulting in x = 2. Signup and view all the answers

How do you solve √x = 6?

Square both sides, resulting in x = 36. Signup and view all the answers

How do you solve ₃√x = 2?

Raise both sides to the third power, resulting in x = 8. Signup and view all the answers

How do you solve r/10 + 4 = 5?

Subtract 4 from both sides and multiply by 10, resulting in r = 10. Signup and view all the answers

How do you solve −9 + n/4 = −7?

Add 9 to both sides and multiply by 4, resulting in n = 8. Signup and view all the answers

How do you solve 1 − r = −5?

Subtract 1 from both sides, resulting in r = 6. Signup and view all the answers

How do you solve x/(−4) − 5 = −8?

Add 5 to both sides, multiply by -4, resulting in x = -12. Signup and view all the answers

What is the solution for x of the equation b - ax = c?

(b - c)/a Signup and view all the answers

Given the equations M = 3 - x and y = N + 4, which of the following expressions is equivalent to M - N written in terms of x and y?

-x - y + 7 Signup and view all the answers

If x and y are positive integers and x - y = 7, what is the least possible value of xy?

8 Signup and view all the answers

For positive real numbers x, y, and z such that 5x = 2y and 1/6y = 2/3z, which of the following inequalities is true?

z < x < y Signup and view all the answers

How do you solve (k - 10) / 2 = -7?

Multiply both sides by 2 and add 10, resulting in k = -4. Signup and view all the answers

Study Notes

Isolating Variables

To isolate a variable, always perform the opposite operation of the number adjacent to it.
For an equation like x - 5 = 2, add 5 to both sides: x = 7.
For x + 6 = 10, subtract 6 from both sides: x = 4.
In 5x = 35, divide both sides by 5 to get x = 7.

Dealing with Fractions

To solve equations with fractions, multiply by the reciprocal.
For (3/5)x = 2, multiply both sides by (5/3) to find x = 10/3.

Working with Exponents

Take the square root to solve x² = 16, yielding x = ±4.
For x³ = 8, use cube roots to find x = 2.
Remember that the square root and square cancel each other out: (√x)² = 6² results in x = 36.
Raise both sides to eliminate the root in ₃√x = 2, resulting in x = 8.

Multi-Step Equations

Apply the PEMDAS method in reverse: handle operations starting with addition or subtraction, then multiplication or division.
In the equation r/10 + 4 = 5, subtract 4 and multiply by 10, leading to r = 10.

Solving for Relationships

For equations involving parameters, manipulate them to find relationships (e.g., x = 1/2z and 3y = 5z) to derive expressions like 10x = 3y.

Understanding Embedded Expressions

Dissolve fractions by multiplying both sides to clear them.
For (k - 10) / 2 = -7, multiply by 2, resulting in k - 10 = -14, leading to k = -4.

Inequalities and Positivity

In inequalities with positive integers (x - y = 7), discern their least product (e.g., x = 8, y = 1).
For constraints involving positive real numbers, analyze the relationships established by given equations to infer inequalities like z < x < y.

Conceptual Equations

Equations can be rewritten or transformed to find equivalent expressions, such as M - N expressed in terms of x and y.

Practice Problems

Engage with various equations and solutions such as n/4 = -7, 1 - r = -5, and check computations involving embedded variables and operations.

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Description

Test your skills in solving basic algebraic equations with these flashcards. Each card presents an equation and walks you through the steps needed to isolate the variable. Perfect for mastering fundamental algebra concepts!