Problem 19 - Elementary Algebra

Question

For the following problems, solve the linear equations in two variables. $$ y+17=x, \text { if } x=-12 $$

Step-by-Step Solution

Answer

Question: Given the linear equation \(y+17=x\) and \(x=-12\), find the value of \(y\). Answer: The value of \(y\) is -29 when \(x = -12\).

Step 1: Substitute the value of x in the equation

Since we know that \(x = -12\), substitute this value to the equation \(y+17=x\): $$ y+17 = -12 $$

Step 2: Solve for y

Now, we will isolate \(y\) by subtracting 17 from both sides of the equation: $$ y = -12 - 17 $$

Step 3: Compute the value of y

Calculate the result: $$ y = -29 $$ So, the solution to the given linear equation is \(y = -29\), when \(x = -12\).

Key Concepts

Two Variables

Linear equations can often involve two variables. These variables are typically represented as "x" and "y". The relationship between these variables is what we aim to understand and solve.
For example, in the equation \( y + 17 = x \), both "y" and "x" are essential parts of the equation. Each variable can influence the outcome of the equation, and they are interconnected in such a way that solving for one usually helps us understand the other.
The goal is to determine the value of one variable when the other is known, allowing us to interpret the relationship between them. In our problem, we know the value of "x", which helps us find "y". Understanding this balance between the two variables is crucial to solving these types of equations efficiently.

Solving Equations

When solving equations, especially those with two variables, your main task is to find the value of the unknowns that satisfy the equation's conditions. This is achieved by isolating one of the variables and finding its corresponding value step-by-step.
The process starts with substituting known values into the equation (if any are available) and simplifying the equation. We then perform operations like addition, subtraction, multiplication, or division to isolate one variable. Each step brings us closer to the solution.
Take our example: after substituting \(x = -12\) into \( y + 17 = x \), we solve for "y" by subtracting 17 from both sides. This step-by-step approach ensures accuracy and clarity in finding the solution.

Substitution Method

The substitution method is a technique used to solve equations, especially useful when dealing with a system of equations. It involves substituting one variable with its known value to simplify and solve the equation for the other variable.
In our problem, we're given \(x = -12\). We substitute "x" in the original equation \(y + 17 = x\). This substitution transforms the equation into \( y + 17 = -12 \), effectively reducing the number of variables we're working with.
By simplifying the equation through substituting the known value, it becomes much easier to solve for the remaining variable. This method is a powerful tool in algebra that streamlines the solution process, making it straightforward to find the values of unknowns.