Solving Equations Comprehensive Guide To Finding X
In the realm of mathematics, solving for x is a fundamental skill that unlocks the door to a multitude of problem-solving capabilities. Whether you're tackling linear equations or delving into the intricacies of quadratic expressions, the ability to isolate x and determine its value is paramount. This comprehensive guide will walk you through the step-by-step process of solving various types of equations, equipping you with the knowledge and confidence to conquer mathematical challenges.
(a) Solving the Linear Equation 25 - 7x = -18
In this section, we'll dissect the linear equation 25 - 7x = -18, providing a meticulous breakdown of each step involved in isolating x and arriving at the solution. Linear equations, characterized by a variable raised to the power of one, are the bedrock of algebraic problem-solving. Mastering their solution is crucial for tackling more complex mathematical concepts.
Step 1 Isolate the Term with x
Our initial objective is to isolate the term containing x, which in this case is -7x. To achieve this, we'll employ the principle of inverse operations. The term 25 is currently added to -7x, so we'll subtract 25 from both sides of the equation. This maintains the equation's balance while effectively moving the constant term to the right side.
25 - 7x - 25 = -18 - 25
This simplifies to:
-7x = -43
Step 2 Solve for x
Now that we have isolated the term with x, we can proceed to solve for x itself. The coefficient of x is -7, indicating that x is being multiplied by -7. To undo this multiplication, we'll divide both sides of the equation by -7. This isolates x on the left side and provides its value on the right side.
-7x / -7 = -43 / -7
This yields the solution:
x = 43/7
Therefore, the exact value of x in the equation 25 - 7x = -18 is 43/7. This can also be expressed as an approximate decimal value of 6.14 (correct to two decimal places).
(b) Solving the Linear Equation 5(x - 4) - (7x - 2) = 2x - 6
Moving on to our next challenge, we encounter a slightly more intricate linear equation: 5(x - 4) - (7x - 2) = 2x - 6. This equation involves parentheses, requiring us to apply the distributive property to simplify the expression before we can isolate x. Let's break down the solution step by step.
Step 1 Distribute and Simplify
The first step in tackling this equation is to eliminate the parentheses. We achieve this by applying the distributive property, which states that a(b + c) = ab + ac. We'll distribute the 5 across the first set of parentheses and the -1 (implicit) across the second set of parentheses.
5(x - 4) = 5x - 20
-(7x - 2) = -7x + 2
Substituting these back into the original equation, we get:
5x - 20 - 7x + 2 = 2x - 6
Now, we'll combine like terms on the left side of the equation.
(5x - 7x) + (-20 + 2) = 2x - 6
This simplifies to:
-2x - 18 = 2x - 6
Step 2 Isolate the x Terms
Next, we want to gather all the terms containing x on one side of the equation. To do this, we'll add 2x to both sides of the equation. This eliminates the -2x term on the left side and moves the x terms to the right side.
-2x - 18 + 2x = 2x - 6 + 2x
This simplifies to:
-18 = 4x - 6
Step 3 Isolate the Constant Terms
Now, we'll isolate the constant terms on the left side of the equation. To do this, we'll add 6 to both sides of the equation. This eliminates the -6 term on the right side and moves the constant terms to the left side.
-18 + 6 = 4x - 6 + 6
This simplifies to:
-12 = 4x
Step 4 Solve for x
Finally, we can solve for x by dividing both sides of the equation by 4. This isolates x on the right side and provides its value on the left side.
-12 / 4 = 4x / 4
This yields the solution:
x = -3
Therefore, the value of x in the equation 5(x - 4) - (7x - 2) = 2x - 6 is -3.
(c) Solving the Linear Equation (3x - 2)/4 = (x + 2)/2 + 1
Our third equation presents us with fractions: (3x - 2)/4 = (x + 2)/2 + 1. To simplify this equation, we'll first eliminate the fractions by multiplying both sides by the least common multiple (LCM) of the denominators. This clears the fractions and allows us to proceed with solving for x.
Step 1 Eliminate Fractions
The denominators in our equation are 4 and 2. The least common multiple of 4 and 2 is 4. Therefore, we'll multiply both sides of the equation by 4.
4 * [(3x - 2)/4] = 4 * [(x + 2)/2 + 1]
This simplifies to:
3x - 2 = 2(x + 2) + 4
Step 2 Distribute and Simplify
Now, we'll distribute the 2 across the parentheses on the right side of the equation.
2(x + 2) = 2x + 4
Substituting this back into the equation, we get:
3x - 2 = 2x + 4 + 4
Combining like terms on the right side, we have:
3x - 2 = 2x + 8
Step 3 Isolate the x Terms
Next, we'll isolate the terms containing x on one side of the equation. To do this, we'll subtract 2x from both sides of the equation.
3x - 2 - 2x = 2x + 8 - 2x
This simplifies to:
x - 2 = 8
Step 4 Solve for x
Finally, we can solve for x by adding 2 to both sides of the equation.
x - 2 + 2 = 8 + 2
This yields the solution:
x = 10
Therefore, the value of x in the equation (3x - 2)/4 = (x + 2)/2 + 1 is 10.
(d) Solving the Quadratic Equation 1.8x² - 3.2x - 7.4 = 0
Our final equation is a quadratic equation: 1.8x² - 3.2x - 7.4 = 0. Quadratic equations, characterized by a variable raised to the power of two, require different solution methods than linear equations. In this case, we'll employ the quadratic formula, a powerful tool for finding the roots of any quadratic equation in the form ax² + bx + c = 0.
Step 1 Identify Coefficients
Before we can apply the quadratic formula, we need to identify the coefficients a, b, and c in our equation. In the equation 1.8x² - 3.2x - 7.4 = 0:
- a = 1.8
- b = -3.2
- c = -7.4
Step 2 Apply the Quadratic Formula
The quadratic formula is given by:
x = [-b ± √(b² - 4ac)] / (2a)
Now, we'll substitute the values of a, b, and c into the formula:
x = [-(-3.2) ± √((-3.2)² - 4 * 1.8 * -7.4)] / (2 * 1.8)
Step 3 Simplify
Let's simplify the expression step by step.
First, we'll calculate the value inside the square root:
(-3.2)² = 10.24
4 * 1.8 * -7.4 = -53.28
10.24 - (-53.28) = 10.24 + 53.28 = 63.52
Now, we'll substitute this back into the equation:
x = [3.2 ± √63.52] / 3.6
Next, we'll calculate the square root of 63.52, which is approximately 7.97 (correct to two decimal places).
x = [3.2 ± 7.97] / 3.6
Step 4 Calculate the Two Solutions
The ± sign indicates that we have two possible solutions. We'll calculate each solution separately.
For the positive case:
x = (3.2 + 7.97) / 3.6
x = 11.17 / 3.6
x ≈ 3.10
For the negative case:
x = (3.2 - 7.97) / 3.6
x = -4.77 / 3.6
x ≈ -1.33
Therefore, the solutions for the quadratic equation 1.8x² - 3.2x - 7.4 = 0 are approximately x = 3.10 and x = -1.33 (correct to two decimal places).
In this comprehensive guide, we've explored various techniques for solving for x in both linear and quadratic equations. We've meticulously dissected each step, providing clear explanations and examples to solidify your understanding. Mastering these techniques is not only essential for academic success but also for tackling real-world problems that require mathematical solutions. Remember, practice is key to honing your problem-solving skills. So, dive into more equations, challenge yourself, and watch your mathematical prowess flourish.
