91Ó°ÊÓ

In algebra, a system of equations is a set of two or more equations involving the same set of variables. Our goal is to find the values of these variables that satisfy all the equations simultaneously. Systems can be solved using various methods, such as substitution, elimination, graphing, and matrix operations.

For the system given in the exercise, we have two equations: \( x = y^2 \) and \( x = 4 \). This system is quite simple, because we can observe that both equations are already solved for the variable \(x\). Therefore, we can directly equate the right-hand sides of both equations—the concept being that if \(a = b\) and \(a = c\), then it must be true that \(b = c\). This approach is a variation of the substitution method and leads us to our Step 1, setting \( y^2 = 4 \).

The square root of a number is a value that, when multiplied by itself, gives the original number. Every positive real number has two square roots: one positive and one negative. This is because both the positive and negative values will square to the original positive number.

In Step 2 of our problem, we encounter the equation \( y^2 = 4 \), which is an example where we need to find the square root of the number 4. Mathematically, we express the solution as \( y = \sqrt{4} \) and \( y = -\sqrt{4} \), yielding two possible values for \(y\). This step is crucial in solving quadratic equations because it points out that such equations may have multiple solutions (in this case, two solutions).

The quadratic formula is a solution to the quadratic equation, \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants. The formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
It offers a systematic way to find the roots of any quadratic equation, using the coefficients of the equation. Although the exercise at hand did not require the use of the quadratic formula, it is instrumental in understanding why we obtain two solutions when solving quadratic equations, as illustrated in Step 2 and Step 3.

The equation from our exercise, \( y^2 = 4 \), is a simple form of a quadratic equation where \(a = 1\), \(b = 0\), and \(c = -4\). Using the quadratic formula would yield the same solutions we calculated by taking the square root. Although it might seem excessive for such a straightforward equation, for more complex quadratics, the quadratic formula is an essential tool.