Problem 60 - Algebra 1: Concepts and Skills

Question

Solve the equation. Round the result to the nearest hundredth. Check the rounded solution. $$-7 x-7=-6$$

Step-by-Step Solution

Answer

-0.14

Step 1: Simplify the Equation

We start with the given equation \$-7x - 7 = -6\$. To simplify, we add 7 to both sides of the equation. This will get rid of the -7 on the left side of the equation, leading to \$-7x = 1\$. From this, we can then move to the next step to isolate 'x'.

Step 2: Solve for 'x'

To solve for 'x', we divide both sides of the equation \`-7x = 1\` by -7. This yields \(x = -\frac{1}{7}\). However, the question asks for the solution to be rounded to the nearest hundredth, hence we compute this as \(x = -0.14\).

Step 3: Verify the Solution

To verify the solution, we substitute \(x = -0.14\) back into the original equation. Doing this gives us \(-7(-0.14) - 7 = -6.02\). This is approximately equal to -6 when rounded to the nearest integer, so our solution is valid when we check the rounded solution.

Key Concepts

Rounding Numbers

Rounding numbers is an essential math skill, applicable in various situations, including solving equations. In this context, rounding helps simplify the solution, making it more manageable for interpretation or application.

When rounding, we focus on a particular decimal place that we want to retain, discarding the rest of the digits after this place. For example, if we need to round a number to the nearest hundredth, we look at the thousandth place:

• If the digit in the thousandth place is 5 or more, we increase the hundredth by one.
• If it's less than 5, we leave the hundredth place as it is.

To apply this to our exercise, the result of dividing -1 by 7 gives us approximately -0.142857. Looking at the thousandth place (which is 2), we round the number to -0.14. Understanding rounding helps in checking and presenting practical solutions.

Checking Solutions

After solving an equation, it's crucial to verify the accuracy of the solution. This step ensures that any rounding errors or calculation mistakes haven't led to an incorrect answer.

Here's a simple approach to checking solutions effectively:

• Substitute the rounded solution back into the original equation.
• Perform the arithmetic operations to determine if the equation holds true.

In our exercise, substituting \(-0.14\) back into \(-7x - 7 = -6\) resulted in \(-6.02\). Rounding \(-6.02\) to the nearest integer yields \(-6\). Since this corresponds with the equation's right-hand side, the original rounded solution checks out. This process confirms that the solution is both reasonable and accurate.

Equation Simplification

Equation simplification is an essential step in solving linear equations as it streamlines the problem. The goal of simplification is to isolate the variable, making it easier to solve.

The basic steps to simplify an equation are:

• Identify and perform arithmetic operations that remove constants from one side.
• Reorganize the equation by applying inverse operations.
• Maintain the equation's balance by performing identical operations on both sides.

In our example, starting with \(-7x - 7 = -6\), we added 7 to each side, simplifying it to \(-7x = 1\). This step removed the constant and brought us closer to isolating \(x\). Finally, dividing by -7 simplified the equation further, isolating \(x\) and facilitating an accurate and precise solution. Grasping simplification is key to tackling any algebra problem effectively.