Question

Prep Exercise 8 For the following equation, complete the related division equation used to solve the equation. \(3 x=-15\) \(x=\) ____ solve and check. $$-1(-1)(-7) g=-63$$

Short answer

Answer: The values of x and g are -5 and -9, respectively.

Step-by-step solution

Solve the first equation, 3x = -15.

To solve this equation, we need to get x alone on one side of the equation. Since 3x means 3 times x, we will perform the inverse operation, which is division. Divide both sides of the equation by 3 to isolate x: \(\frac{3x}{3} = \frac{-15}{3}\) \(x = -5\)

Check the solution for the first equation.

To check our solution, substitute -5 back into the original equation and see if it is true: \(3x = -15\) \(3(-5) = -15\) \(-15 = -15\) Our solution is correct.

Solve the second equation, -1(-1)(-7)g = -63.

First, simplify the left side of the equation by multiplying the factors: \(-1 \times -1 \times -7 \times g = -63\) \(7g = -63\) Now, divide both sides of the equation by 7 to isolate g: \(\frac{7g}{7} = \frac{-63}{7}\) \(g = -9\)

Check the solution for the second equation.

To check our solution, substitute -9 back into the original equation and see if it is true: \(-1(-1)(-7)g = -63\) \(-1(-1)(-7)(-9) = -63\) \(7 \times -9 = -63\) \(-63 = -63\) Our solution is correct.

Key concepts

Inverse Operations

When solving linear equations, the concept of inverse operations plays a crucial role. Inverse operations are mathematical actions that undo each other, such as addition and subtraction or multiplication and division. For instance, if you have a multiplication operation in your equation, as in the expression \(3x\), the inverse operation to isolate the variable \(x\) would be division.

Here's why inverse operations work: If \(3x = -15\), you must perform the same action on both sides of the equation to maintain equality. By dividing each side by 3, you use the inverse operation to clear the coefficient of \(x\), effectively undoing the multiplication. This gives you \(\frac{3x}{3} = \frac{-15}{3}\), which simplifies to \(x = -5\). Always remember, for every action in arithmetic, there is an equal and opposite action, and this principle forms the foundation for solving equations.

Equation Checking

Even after finding a solution for an equation, it's necessary to verify whether the solution is correct or not. This process is known as equation checking and involves substituting the solution back into the original equation to confirm the equality holds true.

To illustrate, after solving \(3x = -15\) and getting \(x = -5\), you'd check the solution by replacing \(x\) with -5 and seeing if the equation balances: \(3(-5) = -15\), which simplifies to \( -15 = -15\). This confirms that \(x = -5\) is indeed the correct solution. The check for the second equation \(\-1(-1)(-7) g = -63\) is similar; you replace \(g\) with -9 and ensure the equation is still balanced. A successful check adds confidence to your solution and demonstrates your understanding of the equation.

Isolating Variables

The goal of solving linear equations is to find the value of an unknown variable. To do this, you must isolate the variable on one side of the equation. Isolating the variable involves using inverse operations to get the variable by itself.

Take the equation \(7g = -63\). To isolate \(g\), you'd divide both sides by 7 (the inverse of multiplying by 7). Similarly, with \(3 x=-15\), you isolated \(x\) by dividing both sides by 3. These steps demonstrate a systematic approach: undo whatever is being done to the variable by performing the opposite action. This strategy ensures that you simplify the equation step by step, moving towards a simple expression where the variable equals a number.