In mathematics, an equation is a statement that asserts the equality of two expressions. A solution of an equation is a value or set of values that, when substituted for the variable(s) in the equation, make the equation true. Let’s break this down with some clear examples.
Understanding Solutions with Simple Equations
Consider the simple equation:
$x + 3 = 5$
To find the solution, we need to determine the value of $x$ that makes this equation true. Subtracting 3 from both sides, we get:
$x = 5 – 3$
$x = 2$
Here, $x = 2$ is the solution because substituting 2 for $x$ in the original equation gives us $2 + 3 = 5$, which is a true statement.
Solutions in More Complex Equations
Equations can become more complex, involving multiple variables or higher powers. For instance, consider the quadratic equation:
$x^2 – 4x + 4 = 0$
To solve this, we can factorize it as:
$(x – 2)(x – 2) = 0$
This gives us the solution $x = 2$. In this case, $x = 2$ is a double root, meaning it satisfies the equation twice.
Systems of Equations
In systems of equations, we are looking for a set of values that satisfy all equations simultaneously. For example:
$2x + y = 10$
$x – y = 2$
We can solve this system using various methods like substitution or elimination. Using substitution, we solve the second equation for $x$:
$x = y + 2$
Substituting this into the first equation, we get:
$2(y + 2) + y = 10$
$2y + 4 + y = 10$
$3y + 4 = 10$
$3y = 6$
$y = 2$
Then substituting $y = 2$ back into $x = y + 2$, we get:
$x = 2 + 2$
$x = 4$
So, the solution to this system is $x = 4$ and $y = 2$
Conclusion
A solution of an equation is essentially the answer that makes the equation true. Whether dealing with simple linear equations, quadratic equations, or more complex systems, finding the solution involves manipulating the equation to isolate the variable(s) and determine their value(s). Understanding how to solve equations is a fundamental skill in mathematics that has wide applications in science, engineering, and everyday problem-solving.