Problem 56 - Algebra 2

Question

Solve each equation. $$ \sqrt{7 x+1}-\sqrt{6 x+7}=0 $$

Step-by-Step Solution

Answer

So, the solution to the equation \(\sqrt{7x + 1} - \sqrt{6x + 7} = 0\) is \(x = 6\).

Step 1: Isolate one of the square roots

Rearrange the given equation to isolate one of the square roots on one side of equation. To do this, add \(\sqrt{6x + 7}\) to both sides of the equation:\[\sqrt{7x + 1} = \sqrt{6x + 7}\]

Step 2: Square both sides of the equation

To remove the square roots, square both sides of the equation:\[7x + 1 = 6x + 7\]

Step 3: Simplify and Solve

Simplify the equation and solve for x by subtracting 6x from both sides to get:\[x = 6\]

Key Concepts

Square Roots

Square roots can initially seem a bit daunting, but they are all about finding a number that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because \(3 \times 3 = 9\). In equations, square roots require careful handling as they can lead to two potential solutions. For instance, \(\sqrt{9}\) can be both 3 and -3, since \((-3) \times (-3) = 9\) as well.
When solving equations involving square roots, it's essential to manipulate the equation to eliminate them, making it easier to solve for the variable. This usually involves isolating the square root on one side so we can square both sides later.

Isolation of Terms

Isolation of terms means arranging the equation so one of the terms stands alone on one side. This step is vital in solving equations, especially those with square roots, to simplify subsequent operations. For instance, in the equation \(\sqrt{7x + 1} - \sqrt{6x + 7} = 0\), the first step is to move one of the square roots across the equation.
This would look like:

• Add \(\sqrt{6x + 7}\) to both sides to maintain equality.
• You will get \(\sqrt{7x + 1} = \sqrt{6x + 7}\).

By isolating a square root term in this way, we prepare the equation for the next critical step: squaring.

Squaring Both Sides

Squaring both sides of an equation is a powerful technique to eliminate square roots, making the equation solvable using simpler algebra. Using square roots can often lead to simpler expressions that are much easier to solve.
Following the isolation of a square root, squaring both sides involves:

• Taking the equation \(\sqrt{7x + 1} = \sqrt{6x + 7}\).
• Squaring both sides to get rid of the roots, resulting in \(7x + 1 = 6x + 7\).

Once the roots are eliminated, the equation transforms into a more straightforward linear equation ready for simplification.

Simplifying Equations

Simplifying an equation means breaking it down to its simplest form to find the solution easily. This often involves basic algebraic operations like addition, subtraction, multiplication, or division.
After squaring the sides in our problem, our equation simplifies to:

• \(7x + 1 = 6x + 7\).
• Subtract \(6x\) from both sides to find what remains on both sides of the equation: \(x + 1 = 7\).
• Solving gives \(x = 6\) by subtracting 1 from both sides.

This final step reveals the solution to the equation, confirming the value of the variable.