Substitution is the process of replacing one variable or expression with another equivalent expression in an equation or expression. This technique is used in various mathematical contexts to simplify, solve, or manipulate equations and expressions.
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Substitution is a key technique used in algebra to simplify expressions, solve equations, and work with systems of equations.
When multiplying or dividing integers, substitution can be used to rewrite expressions in a more manageable form.
In problem-solving strategies, substitution can be employed to represent unknown quantities with variables and then solve for those variables.
Substitution is a crucial step in solving systems of equations by graphing, as it allows you to eliminate variables and find the point of intersection.
Substitution is often used in conjunction with other algebraic techniques, such as combining like terms, factoring, and using the distributive property.
Review Questions
How can substitution be used to simplify expressions when using the language of algebra?
When working with the language of algebra, substitution can be used to replace variables or expressions with equivalent expressions to simplify the overall expression. For example, if you have the expression $3x + 2y$ and you know that $x = 4$ and $y = 2$, you can substitute these values to simplify the expression to $3(4) + 2(2) = 12 + 4 = 16$.
Explain how substitution can be used to multiply and divide integers.
Substitution can be a helpful technique when multiplying and dividing integers. For instance, if you have the expression $-3 \cdot (-5)$, you can substitute the variables $a = -3$ and $b = -5$ to rewrite the expression as $a \cdot b$. This allows you to apply the rules for multiplying integers, in this case, that a negative times a negative results in a positive. So, the final answer would be $-3 \cdot (-5) = 15$.
Analyze how substitution is used as a problem-solving strategy.
Substitution is a crucial problem-solving strategy, as it allows you to represent unknown quantities with variables and then solve for those variables. For example, if you have a word problem that asks you to find the cost of $x$ items at $y$ dollars each, you can let $x$ and $y$ represent the unknown quantities and then use substitution to set up an equation, such as $\text{total cost} = x \cdot y$. By solving for $x$ or $y$, you can determine the missing information and arrive at the final solution.