Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. Equations Having Symbols of Grouping. $$5(3 x-13)=10$$

Short answer

The solution to the equation is \( x = 5 \).

Step-by-step solution

Distribute the Multiplication

Multiply the term outside the parentheses, which is 5, by each term inside the parentheses: $$5(3x - 13) = 5 \times 3x - 5 \times 13$$. This gives us $$15x - 65$$. This step simplifies the left side of the equation.

Isolate the Variable

To solve for x, we need to isolate it on one side of the equation. We can add 65 to both sides to cancel out the -65 on the left: $$15x - 65 + 65 = 10 + 65$$. This simplifies to $$15x = 75$$.

Solve for x

To solve for x, divide both sides of the equation by 15: $$15x / 15 = 75 / 15$$. Simplifying, we get $$x = 5$$.

Check the Solution

Substitute the value of x back into the original equation to check if the left side equals the right side: $$5(3(5) - 13) = 5(15 - 13) = 5(2) = 10$$. Since this is the same as the right side of the original equation, our solution is correct.

Key concepts

Symbols of Grouping

Understanding the symbols of grouping is essential when solving linear equations. These symbols, such as parentheses \( ( ) \) and brackets \[ [ ] \], tell us that we should perform all operations inside them before moving on to operations outside. In the given equation \(5(3x - 13) = 10\), the parentheses around \(3x - 13\) indicate that we should multiply 5 by both \(3x\) and -13.
When distributing, we multiply the number outside the parentheses by each term inside, resulting in \(15x - 65\). This step eliminates the grouping and simplifies the equation, allowing us to move onto the next steps with ease. It's a fundamental skill to apply the distributive property correctly to eliminate the groupings and simplify equations.

Isolating the Variable

To solve an equation, our ultimate goal is to isolate the variable—in this case, \(x\)—on one side of the equation. \(15x - 65\) is simplified further by adding 65 to both sides, neutralizing the -65 and leaving \(15x\) by itself on the left side of the equation.
Isolating the variable is like finding the missing piece of a puzzle. We perform operations such as addition, subtraction, multiplication, or division on both sides of the equation to get the variable alone. In our problem, after we added 65 to both sides, we divided by 15 to isolate \(x\), resulting in the solution \(x = 5\). By isolating \(x\), we've effectively solved the puzzle.

Equation Checking

Checking your solution is a critical step in solving equations. After we've isolated the variable and found \(x = 5\), we need to confirm that it works in the original equation. We do this by substituting \(5\) back into the original equation instead of \(x\) and ensuring that both sides are equal. The process is like reverse-engineering the problem: if we end up with a true statement, our solution is valid.
In our case, we substitute \(x\) with \(5\) in \(5(3x - 13)\) leading to \(5(15 - 13)\) and finally \(5(2)\), which equals 10. Since this matches the right side of the equation, we celebrate—the solution \(x = 5\) is correct!

Fractional Solutions

Sometimes we encounter solutions in the form of fractions instead of whole numbers. These fractional solutions are just as valid and often more accurate than decimal approximations. When solving equations, if you get a fraction as the solution, it's important to simplify it fully but keep it in fractional form for precision.
For example, if we had a result like \(75 / 15\), instead of converting it to a decimal \(5.0\), we simplify the fraction to \(5/1\) or simply \(5\), which is the most accurate representation of the solution. Keeping solutions in fractions ensures the exactness needed in mathematical problems without the need for rounding errors that decimals might introduce.