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 $2x + 3 = 11$
- Isolate the variable: Subtract 3 from both sides:
$2x + 3 – 3 = 11 – 3$
$2x = 8$ - Solve for x: Divide both sides by 2:
$x = frac{8}{2}$
$x = 4$
Example 2: Linear Equation with Fractions
Let’s solve $frac{3x}{4} – 2 = 1$
- Eliminate the fraction: Multiply both sides by 4 to get rid of the denominator:
$4 times frac{3x}{4} – 4 times 2 = 4 times 1$
$3x – 8 = 4$ - Isolate the variable: Add 8 to both sides:
$3x – 8 + 8 = 4 + 8$
$3x = 12$ - Solve for x: Divide both sides by 3:
$x = frac{12}{3}$
$x = 4$
Solving Quadratic Equations
Example 3: Factoring Method
Consider the equation $x^2 – 5x + 6 = 0$
- Factor the quadratic: Find two numbers that multiply to 6 and add to -5. These numbers are -2 and -3:
$(x – 2)(x – 3) = 0$ - Set each factor to zero:
$x – 2 = 0$ or $x – 3 = 0$ - Solve for x:
$x = 2$ or $x = 3$
Example 4: Quadratic Formula
Consider the equation $x^2 + 4x + 4 = 0$
- Identify coefficients: Here, $a = 1$, $b = 4$, and $c = 4$
- 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:
$begin{cases}
2x + y = 10
3x – y = 5
end{cases}$
- Solve one equation for one variable: From the first equation, solve for y:
$y = 10 – 2x$ - Substitute into the second equation:
$3x – (10 – 2x) = 5$
$3x – 10 + 2x = 5$
$5x – 10 = 5$ - Solve for x:
$5x = 15$
$x = 3$ - 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!