Algebra, with its unknown variables and abstract concepts, can be intimidating for many students. However, understanding how to solve for x is a fundamental skill in algebra that forms the basis for more complex equations. Solving for x simply means finding the value of x that makes the equation true. This process involves manipulating the equation to isolate x on one side and simplifying the equation to find the solution.
In this article, we will explore various methods and techniques to solve for x, including solving equations with fractions, in triangles, and with quadratic equations. We will also provide step-by-step instructions and solved examples to help you understand the process better. So, let’s dive into the world of solving for x and unlock the mysteries of algebra!
Solve for X Mean
When we talk about “solving for x” we are referring to finding the value of x that satisfies a given equation. In other words, we are searching for the unknown variable x that makes the equation true. When we find the value of x and substitute it back into the equation, the left-hand side (LHS) should equal the right-hand side (RHS). Solving for x is a crucial step in many mathematical and scientific calculations.
For example, if we have the equation “x + 1 = 2,” solving for x means finding a value for x that satisfies the equation. Let’s try substituting x = 1 into the equation:
1 + 1 = 2
2 = 2
Since the LHS equals the RHS, we can conclude that x = 1 is the solution to the equation.
How To Solve for x?
To solve for x, we need to manipulate the equation to isolate x on one side and simplify the equation to find the solution. The goal is to perform the same operation on both sides of the equation, maintaining equality throughout the process. Here is a step-by-step guide on how to solve for x:
Step 1: Start with a simple equation
Let’s begin with a basic equation to illustrate the process. Consider the equation “x + 2 = 7.”
Step 2: Get x by itself
To isolate x on one side of the equation, we need to move the constant term (2) to the other side. We can achieve this by subtracting 2 from both sides:
x + 2 – 2 = 7 – 2
x = 5
Step 3: Check the answer
To verify our solution, we substitute x = 5 back into the original equation:
5 + 2 = 7
7 = 7
Since the equation holds true, we can conclude that x = 5 is the solution.
Solve for x in the Triangle
Solving for x in a triangle involves using the properties of triangles or the Pythagorean theorem. Let’s explore two scenarios: solving for x to find the missing side length and solving for x to find the missing angle.
Solving for x to find the missing side length of a triangle
Suppose we have a right-angled triangle △ABC, where angle B is 90 degrees. The lengths of the two legs are given as 7 units and 24 units. We want to find the length of the hypotenuse, denoted as x.
By applying the Pythagorean theorem, we have:
AC² = AB² + BC²
x² = 7² + 24²
x² = 49 + 576
x² = 625
Taking the square root of both sides, we get:
x = √625
x = 25 units
Therefore, the length of the hypotenuse is 25 units.
Solving for x to find the missing angle of a triangle
Consider triangle ABC, where angle A is 50 degrees, angle B is 60 degrees, and angle C is denoted as x. We want to find the value of x.
Using the angle sum property of triangles, we know that the sum of the angles in a triangle is 180 degrees. Therefore, we can set up the following equation:
angle A + angle B + angle C = 180 degrees
50° + 60° + x° = 180°
x = 70°
Hence, the value of angle C is 70 degrees.
Solve for x in Fractions
Solving for x in fractions involves cross-multiplication and simplifying the equation to find x. Let’s take an example equation: 2/5 = x/10.
Step 1: Cross-multiply the fractions
2 × 10 = 5 × x
20 = 5x
Step 2: Solve the equation for x
x = 20 / 5
x = 4
To verify our solution, we substitute x = 4 back into the original equation:
2/5 = 4/10
2 × 10 = 5 × 4
20 = 20
Since the equation holds true, we can conclude that x = 4 is the solution.
Solve for x Equations
When dealing with equations that involve different variables, we can use various methods to solve for x. One common approach is the substitution method, which involves solving one equation for x in terms of another variable and substituting it into the second equation. Let’s explore an example to illustrate this method:
Consider the system of equations:
2x – y = 5 (Equation 1) 3x + 2y = 11 (Equation 2)
Step 1: Solve Equation 1 for x
2x – y = 5
2x = 5 + y
x = (5 + y) / 2 (Equation 3)
Step 2: Substitute Equation 3 into Equation 2
3((5 + y) / 2) + 2y = 11
(15 + 3y) / 2 + 2y = 11
(15 + 3y + 4y) / 2 = 11
(15 + 7y) / 2 = 11
15 + 7y = 22
7y = 22 – 15
7y = 7
y = 1
Step 3: Substitute y = 1 into Equation 3
x = (5 + 1) / 2
x = 6 / 2
x = 3
Therefore, the solution to the system of equations is x = 3 and y = 1.
Solved Examples on Solve for x
Let’s work through a few more examples to solidify our understanding of solving for x.
Example 1: Solve for x in the equation 4x + 2 = -8.
Step 1: Get x by itself
Subtract 2 from both sides:
4x + 2 – 2 = -8 – 2
4x = -10
Step 2: Divide both sides by 4
4x/4 = -10/4
x = -5/2
Example 2: Solve for x in the equation 3x – 7 = 26.
Step 1: Get x by itself
Add 7 to both sides:
3x – 7 + 7 = 26 + 7
3x = 33
Step 2: Divide both sides by 3
3x/3 = 33/3
x = 11
Example 3: Solve for x in the equation √(2x+9) – 5 = 0.
Step 1: Get x by itself
Add 5 to both sides:
√(2x+9) – 5 + 5 = 0 + 5
√(2x+9) = 5
Step 2: Square both sides
(√(2x+9))² = 5²
2x + 9 = 25
Step 3: Solve for x
2x = 25 – 9
2x = 16
x = 8
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