Solve This Equation: $\frac{3}{5}(x-10)=18-4x-1$Step 1: Simplify Using The Distributive Property. Which Number Can Be Distributed Across Two Terms Inside The Parentheses?$\frac{3}{5}(x-10)=18-4x-1$

Step 1: Simplify Using the Distributive Property

When solving linear equations, it's essential to simplify the equation using the distributive property. This property allows us to distribute a coefficient across two or more terms inside parentheses. In this case, we have the equation $\frac{3}{5}(x-10)=18-4x-1$. To simplify, we need to distribute the coefficient $\frac{3}{5}$ across the two terms inside the parentheses, which are $x$ and $-10$.

Distributing the Coefficient

The distributive property states that for any real numbers $a$, $b$, and $c$, we have:

$a(b+c) = ab + ac$

In our case, we can apply this property by distributing $\frac{3}{5}$ across $x$ and $-10$:

$\frac{3}{5}(x-10) = \frac{3}{5}x - \frac{3}{5}(10)$

Now, we can simplify the right-hand side of the equation by evaluating the product of $\frac{3}{5}$ and $10$:

$\frac{3}{5}(x-10) = \frac{3}{5}x - 6$

Simplifying the Right-Hand Side

The right-hand side of the equation is $18-4x-1$. We can simplify this expression by combining like terms:

$18-4x-1 = 17-4x$

Now, we have the simplified equation:

$\frac{3}{5}x - 6 = 17-4x$

Combining Like Terms

To solve for $x$, we need to combine like terms on both sides of the equation. We can start by moving all the terms with $x$ to one side of the equation and the constant terms to the other side:

$\frac{3}{5}x + 4x = 17 + 6$

Adding the Coefficients

Now, we can add the coefficients of $x$ on the left-hand side of the equation:

$\frac{3}{5}x + 4x = \frac{3}{5}x + \frac{20}{5}x$

Combining the fractions, we get:

$\frac{3}{5}x + \frac{20}{5}x = \frac{23}{5}x$

Simplifying the Right-Hand Side

The right-hand side of the equation is $17 + 6$. We can simplify this expression by adding the constants:

$17 + 6 = 23$

Now, we have the simplified equation:

$\frac{23}{5}x = 23$

Solving for $x$

To solve for $x$, we can divide both sides of the equation by $\frac{23}{5}$:

$x = \frac{23}{\frac{23}{5}}$

Evaluating the Expression

To evaluate the expression, we can simplify the fraction by canceling out the common factor of $23$:

$x = \frac{23}{\frac{23}{5}} = 5$

Therefore, the solution to the equation is $x = 5$.

Conclusion

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Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. It can be written in the form ax + b = c, where a, b, and c are constants.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to distribute a coefficient across two or more terms inside parentheses. It states that for any real numbers a, b, and c, we have:

a(b+c) = ab + ac

Q: How do I simplify an equation using the distributive property?

A: To simplify an equation using the distributive property, you need to distribute the coefficient across the two or more terms inside parentheses. For example, if you have the equation:

a(x+y) = b

You can simplify it by distributing the coefficient a across the terms x and y:

ax + ay = b

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable (usually x) is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable (usually x) on one side of the equation. You can do this by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. For example, if you have the equation:

2x + 3 = 5

You can solve it by subtracting 3 from both sides:

2x = 5 - 3 2x = 2

Then, you can divide both sides by 2:

x = 2/2 x = 1

Q: What is the order of operations?

A: The order of operations is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The order of operations is:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I check my solution to a linear equation?

A: To check your solution to a linear equation, you need to plug the value of the variable back into the original equation and see if it is true. For example, if you have the equation:

2x + 3 = 5

And you solve it to get x = 1, you can plug x = 1 back into the original equation:

2(1) + 3 = 5 2 + 3 = 5 5 = 5

Since the equation is true, you know that your solution is correct.

Conclusion

Solving linear equations requires careful simplification and manipulation of the equation. By applying the distributive property and combining like terms, you can isolate the variable and solve for its value. Remember to check your solution by plugging the value of the variable back into the original equation.