How to Determine the Value of x?

Determining the value of x is a fundamental skill in algebra and is crucial for solving equations. Let’s explore some common methods to find x.

Solving Linear Equations

Example 1: Simple Linear Equation

Consider the equation $2x + 3 = 11$

1. Isolate the variable: Subtract 3 from both sides:
   $2x + 3 – 3 = 11 – 3$
   $2x = 8$
2. Solve for x: Divide both sides by 2:
   $x = frac{8}{2}$
   $x = 4$

Example 2: Linear Equation with Fractions

Let’s solve $frac{3x}{4} – 2 = 1$

1. Eliminate the fraction: Multiply both sides by 4 to get rid of the denominator:
   $4 times frac{3x}{4} – 4 times 2 = 4 times 1$
   $3x – 8 = 4$
2. Isolate the variable: Add 8 to both sides:
   $3x – 8 + 8 = 4 + 8$
   $3x = 12$
3. Solve for x: Divide both sides by 3:
   $x = frac{12}{3}$
   $x = 4$

Solving Quadratic Equations

Example 3: Factoring Method

Consider the equation $x^2 – 5x + 6 = 0$

1. Factor the quadratic: Find two numbers that multiply to 6 and add to -5. These numbers are -2 and -3:
   $(x – 2)(x – 3) = 0$
2. Set each factor to zero:
   $x – 2 = 0$ or $x – 3 = 0$
3. Solve for x:
   $x = 2$ or $x = 3$

Example 4: Quadratic Formula

Consider the equation $x^2 + 4x + 4 = 0$

1. Identify coefficients: Here, $a = 1$, $b = 4$, and $c = 4$
2. Apply the quadratic formula:
   $x = frac{-b , pm , sqrt{b^2 – 4ac}}{2a}$
   Plugging in the values:
   $x = frac{-4 , pm , sqrt{4^2 – 4 cdot 1 cdot 4}}{2 cdot 1}$
   $x = frac{-4 , pm , sqrt{16 – 16}}{2}$
   $x = frac{-4 , pm , 0}{2}$
   $x = frac{-4}{2}$
   $x = -2$

Solving Systems of Equations

Example 5: Substitution Method

Consider the system:
$begin{cases}
2x + y = 10
3x – y = 5
end{cases}$

1. Solve one equation for one variable: From the first equation, solve for y:
   $y = 10 – 2x$
2. Substitute into the second equation:
   $3x – (10 – 2x) = 5$
   $3x – 10 + 2x = 5$
   $5x – 10 = 5$
3. Solve for x:
   $5x = 15$
   $x = 3$
4. Find y: Substitute $x = 3$ back into $y = 10 – 2x$:
   $y = 10 – 2(3)$
   $y = 4$

Conclusion

By mastering these methods, you can solve for x in various types of equations. Practice makes perfect, so keep working on different problems to strengthen your skills!

Citations

1. 1. Khan Academy – Solving Equations
2. 2. Purplemath – Solving Quadratic Equations
3. 3. Math is Fun – Systems of Equations