Problem 86 - Elementary Algebra

Question

Solve \(-3 y+10=x+2\) if \(x=-4\)

Step-by-Step Solution

Answer

Answer: The value of \(y\) when \(x=-4\) is \(y=4\).

Step 1: Substitute the value of \(x\) into the equation

Replace \(x\) with \(-4\) in the equation \(-3 y+10=x+2\): $$ -3y + 10 = -4 + 2 $$

Step 2: Simplify the equation

Perform the calculation on the right side of the equation: $$ -3y + 10 = -2 $$

Step 3: Isolate the term with \(y\)

To isolate \(y\), we can subtract \(10\) from both sides of the equation: $$ -3y = -2 - 10 $$

Step 4: Simplify the equation

Perform the calculation on the right side of the equation: $$ -3y = -12 $$

Step 5: Solve for y

Divide both sides of the equation by \(-3\) to solve for \(y\): $$ y = \frac{-12}{-3} $$

Step 6: Final answer

Perform the calculation to get the final answer: $$ y = 4 $$

Key Concepts

Substituting Values

Understanding how to substitute values is a fundamental skill in solving linear equations. When a variable's value is provided, as in the case where we have a known value for x, substituting that value into the equation helps to simplify it. This process involves replacing the variable with its corresponding numerical value.

For example, consider the equation from the exercise: \( -3y + 10 = x + 2 \). If we know that x equals -4, then we substitute \( x \) with \( -4 \) which gives us the new equation \( -3y + 10 = -4 + 2 \). Substituting correctly is crucial because it sets the stage for the following steps needed to solve for the unknown variable.

Isolate Variable

To find the value of the variable, such as y in our equation, we need to isolate the variable. This means we want the variable on one side of the equation with everything else on the other side. To do this, we can use operations like addition or subtraction to remove any constants from the side of the variable, and multiplication or division to remove any coefficients.

Continuing with our equation, \( -3y = -12 \), we want to get y by itself. As y is being multiplied by -3, we isolate y by doing the opposite operation: division. When we divide both sides by -3, we are applying the principle of balance—that is, what we do to one side, we must do to the other to maintain equality. The result is \( y = \frac{-12}{-3} \) which simplifies to \( y = 4 \).

Simplifying Equations

Simplifying equations reduces them to their simplest form, making them easier to solve. This involves performing arithmetic operations such as addition, subtraction, multiplication, and division to combine like terms and eliminate parentheses. When we simplified the equation \( -3y + 10 = -4 + 2 \) by carrying out the addition on the right side, we got a simpler equation: \( -3y + 10 = -2 \).

Later, after isolating the variable, we further simplified \( -3y = -2 - 10 \) to \( -3y = -12 \). Simplifying each step systematically helps to ensure that no mistakes are made, and it makes the solving process more straightforward.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the exercise, the equation \( -3y + 10 = x + 2 \) is an algebraic expression that represents a balance of two sides. Understanding how to manipulate these expressions is key to solving equations.

An important part of working with algebraic expressions is the use of properties such as the distributive property, the associative property, and the commutative property. These properties allow for the rearrangement and combination of terms that facilitate simplification and eventually lead us to find the value of the variables involved.