Question

Evaluate the expression when $x=3,y=-2$, and $z=4\$\$x+3y+z$

Short answer

The value of the expression when $x=3,y=-2$, and $z=4$ is $1$.

Step-by-step solution

Substitute the Values

Firstly, replace each variable in the expression with the given values. Hence, replace $x$ with $3$, $y$ with $-2$, and $z$ with $4$ in the expression $x+3y+z$.

Evaluate the Expression

After the substitution, the expression is $3+3(-2)+4$. Now, perform the multiplication first as per the BODMAS rule (Brackets, Orders, Division & Multiplication, Addition & Subtraction). Calculate $3\ast (-2)$ which is $-6$. Next perform the addition: $3-6+4$.

Simplify the Expression

Finally, simplify the expression to find the final value. $3-6+4$ becomes $-3+4$, which further simplifies to $1$.

Key concepts

Substitution Method in Algebra

In algebra, the substitution method is a fundamental process used to evaluate expressions. This method involves replacing variables with their given numerical values.

When you come across a problem that asks you to evaluate the expression such as the one provided where you need to find the value when ($x=3,y=-2,z=4$), this is effectively what you'd do:

• Look at the original expression, in this case, ($x+3y+z$).
• Replace each variable with the corresponding numbers given, so ($x$) becomes ($3$), ($y$) becomes ($-2$), and ($z$) becomes ($4$).
• The new expression, after substitution, should appear as ($3+3(-2)+4$).

Remember that you should not change anything else about the expression while substituting; just swap the letters for numbers and maintain the original operation signs.

Order of Operations

A common method used to remember the order of operations in algebra is BODMAS or PEMDAS (Parentheses, Exponents, Multiplication & Division, Addition & Subtraction).

This rule helps us decide which part of the expression to calculate first. After replacing the variables using the substitution method:

• Look inside parentheses, if any, and solve the operations there first.
• Next, tackle any exponents.
• Then, perform multiplication and division from left to right.
• Finally, carry out addition and subtraction, again moving from left to right.

In our example, you multiply first, calculating ($3(-2)$) which gives ($-6$). The expression now becomes ($3-6+4$). No parentheses or exponents are present, so you simply complete the addition and subtraction in sequence to arrive at the answer.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and at least one arithmetic operation.

When constructing or interpreting an algebraic expression, it's essential to recognize that:

• Variables, represented by letters like ($x,y,z$), stand for unknown values or values that can change.
• Coefficients are the numbers that multiply a variable, like the ($3$) in ($3y$), which indicates that the variable ($y$) should be multiplied by ($3$).
• Constants are fixed numbers that don’t change, such as ($4$) in our given expression.
• Operation symbols (+, -, \times, \div) tell you how to combine these elements.

Evaluating an algebraic expression means finding its numerical value. For the provided exercise, once you've performed the substitution and applied the order of operations, you should be left with a simplified numerical value that answers the question posed.