Problem 42
Question
evaluate each algebraic expression for the given value of the variable or variables. $$ 6(x+5)-13 ; x=-7 $$
Step-by-Step Solution
Verified Answer
-25
1Step 1: Substitute for variable
Substitute the given value of \( x = -7 \) into the equation: \( 6((-7)+5) - 13 \)
2Step 2: Simplify inside parentheses
-7 + 5 equals -2, so we get: \( 6*(-2) - 13 \)
3Step 3: Perform multiplication
6 multiplied by -2 equals -12, so now the equation is: \( -12 - 13 \)
4Step 4: Perform subtraction
-12 minus 13 equals -25
Key Concepts
Substitution MethodSimplifying ExpressionsArithmetic Operations in Algebra
Substitution Method
When we evaluate algebraic expressions, the substitution method is our first step to solving the problem. This involves replacing variables with actual numbers given in the exercise.
To put it simply, imagine each variable as an empty box, and your job is to fill that box with the given value. For instance, if you have the expression \(6(x+5)-13\) and you're told that \(x=-7\), you'll place '-7' where 'x' is in the expression. After substitution, what you have is \(6((-7)+5)-13\), which looks a bit less daunting! This initial switch is crucial because it sets the stage for all the following steps.
The key reason this method is so helpful is that it transforms abstract expressions into concrete numbers that we can more easily handle with basic arithmetic operations.
To put it simply, imagine each variable as an empty box, and your job is to fill that box with the given value. For instance, if you have the expression \(6(x+5)-13\) and you're told that \(x=-7\), you'll place '-7' where 'x' is in the expression. After substitution, what you have is \(6((-7)+5)-13\), which looks a bit less daunting! This initial switch is crucial because it sets the stage for all the following steps.
The key reason this method is so helpful is that it transforms abstract expressions into concrete numbers that we can more easily handle with basic arithmetic operations.
Simplifying Expressions
Once we've replaced our variables, it's time to simplify the expression. This process is about making our math as straightforward as possible. We should first look inside parentheses and perform any arithmetic operations there.
In our example, we'll simplify inside the parentheses first. So, \((-7)+5\) becomes \(-2\). This step is often missed, but it's essential for reducing errors in our calculations.
After addressing the parentheses, you can apply the 'order of operations' to continue simplifying. This might involve multiplying or dividing numbers first, then moving on to addition and subtraction. Being methodical ensures that nothing is overlooked and that you get the correct answer at the end of your simplification journey.
In our example, we'll simplify inside the parentheses first. So, \((-7)+5\) becomes \(-2\). This step is often missed, but it's essential for reducing errors in our calculations.
After addressing the parentheses, you can apply the 'order of operations' to continue simplifying. This might involve multiplying or dividing numbers first, then moving on to addition and subtraction. Being methodical ensures that nothing is overlooked and that you get the correct answer at the end of your simplification journey.
Arithmetic Operations in Algebra
Algebra is not a mysterious language; it's just regular arithmetic dressed up with some letters! Once we've substituted and simplified, we proceed to the arithmetic operations we've known since grade school—addition, subtraction, multiplication, and division.
Look at our example: We have \(6*(-2)-13\) left after simplifying. Here, we do the multiplication first, following the order of operations—multiplication before subtraction. This gives us \(-12-13\). Finally, it's just a bit of subtraction left, and voilà! We arrive at our answer, \(-25\).
It's remarkable how a tangled web of algebraic expressions can be untangled through basic arithmetic operations. And remember, always keep the order of operations in mind, as it will guide you to the correct answer without fail.
Look at our example: We have \(6*(-2)-13\) left after simplifying. Here, we do the multiplication first, following the order of operations—multiplication before subtraction. This gives us \(-12-13\). Finally, it's just a bit of subtraction left, and voilà! We arrive at our answer, \(-25\).
It's remarkable how a tangled web of algebraic expressions can be untangled through basic arithmetic operations. And remember, always keep the order of operations in mind, as it will guide you to the correct answer without fail.
Other exercises in this chapter
Problem 41
Add or subtract as indicated. $$ \frac{3}{x+4}+\frac{6}{x+5} $$
View solution Problem 42
Simplify each exponential expression $$ \left(-\frac{6}{y}\right)^{3} $$
View solution Problem 42
Find each product. $$(x+5)^{2}$$
View solution Problem 42
In Exercises \(41-48,\) factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}+4 x+4$$
View solution