Here is a linear equation in two variables: 2x + 4y – 31 = 123. Solve the equation, first for x and then for y.
Here is a linear equation in two variables: 2x + 4y – 31 = 123. Solve the equation, first for x and then for y.
The Correct Answer and Explanation is:
To solve the linear equation
2x + 4y – 31 = 123,
we first simplify the equation and then solve for one variable in terms of the other.
Step 1: Simplify the equation
Start by moving constants to one side:
2x + 4y – 31 = 123
Add 31 to both sides:
2x + 4y = 154
Step 2: Solve for x in terms of y
Now, isolate x:
2x + 4y = 154
Subtract 4y from both sides:
2x = 154 – 4y
Divide both sides by 2:
x = 77 – 2y
Step 3: Solve for y in terms of x
Go back to the simplified equation:
2x + 4y = 154
Subtract 2x from both sides:
4y = 154 – 2x
Divide both sides by 4:
y = (154 – 2x)/4
Or simplify:
y = 38.5 – 0.5x
Final Answers:
- x = 77 – 2y
- y = 38.5 – 0.5x
Explanation
A linear equation in two variables, such as 2x + 4y – 31 = 123, represents a straight line in the coordinate plane. The solution set of this equation is not a single point but an infinite number of ordered pairs (x, y) that satisfy the relationship. The goal is to express one variable in terms of the other to better understand the relationship between them.
We begin by simplifying the equation to isolate the variables and constants clearly. Adding 31 to both sides removes the constant on the left, yielding 2x + 4y = 154. This is now in a cleaner form to manipulate algebraically.
To solve for x, we isolate it by moving the 4y to the other side, resulting in 2x = 154 – 4y, and then divide both sides by 2. This gives us x = 77 – 2y, which shows that x depends on the value of y. This form is useful when you know y and want to compute x.
Similarly, we can solve for y in terms of x. Subtracting 2x and dividing by 4 gives y = 38.5 – 0.5x. This allows you to calculate y if you know x.
These forms also help when graphing the equation or finding intercepts. For instance, setting y = 0 in the first equation yields x = 77, giving the x-intercept. This method is foundational in algebra for handling linear systems, substitutions, and graphing.
