What is the Y-Value of the Solution to the System of Equations? 3x + 5y = 1 7x + 4y = −13 −3 −1 2 5
When solving a system of equations, finding the y-value of the solution is crucial to understanding the relationship between the variables. In this case, we are given two equations: 3x + 5y = 1 and 7x + 4y = -13. The goal is to determine the value of y that satisfies both equations simultaneously.
To find the y-value, we can use methods such as substitution or elimination. By rearranging one of the equations and substituting it into the other, we can isolate and solve for y. Alternatively, by multiplying one equation by a suitable constant and adding or subtracting it from the other equation, we can eliminate one variable and solve for the remaining variable.
Understanding the System of Equations
When faced with a system of equations, like the one given: 3x + 5y = 1 and 7x + 4y = -13, it’s important to comprehend the underlying concepts to determine the y-value of the solution. Solving this system involves finding values for both x and y that satisfy both equations simultaneously.
To begin, we can use various methods such as substitution or elimination to find the solution. Let’s explore how substitution works in this case:
Step 1: Solve one equation for either variable. From the first equation, we can isolate x by subtracting 5y from both sides: 3x = 1 – 5y
Step 2: Substitute the expression obtained into the other equation. Replacing x in the second equation with (1 – 5y), we get: 7(1 – 5y) + 4y = -13
Step 3: Simplify and solve for y. Expanding and rearranging terms, we have: 7 -35y +4y = -13 -31y = -20 y ≈ (-20)/(-31) y ≈ (20/31)
Therefore, after evaluating y using substitution, we find that approximately y is equal to (20/31).
It’s crucial to note that there are alternative methods like elimination or graphing available to solve systems of equations. Each method may yield slightly different results due to rounding errors or approximation techniques used during calculations.
Understanding how to approach systems of equations empowers us to find solutions accurately. By applying appropriate techniques like substitution or elimination, we can calculate specific values such as the y-value in relation to given equations. Solving the First Equation
Let’s dive into solving the first equation of the system: 3x + 5y = 1. Our goal is to find the value of y that satisfies this equation. To do so, we’ll apply some algebraic techniques.
We start by isolating y on one side of the equation. Subtracting 3x from both sides gives us 5y = 1 – 3x. Now, to solve for y, we need to divide both sides by 5: y = (1 – 3x)/5.
The resulting expression provides us with a formula for calculating the y-value based on any given x-value. For example, if we substitute x = -3 into the equation, we can determine its corresponding y-value as follows:
y = (1 – 3(-3))/5 = (1 +9)/5 =10/5 =2
So when x equals -3, our solution tells us that y equals 2.
Solving the Second Equation
Now let’s dive into solving the second equation of the system: 7x + 4y = -13. This equation, along with the first equation (3x + 5y = 1), forms a system that can be solved to find the values of x and y.
To solve this equation, we’ll use a method called substitution. The idea behind substitution is to isolate one variable in terms of the other from one equation and then substitute it into the other equation. In this case, we’ll solve for x in terms of y using the first equation.
Starting with our first equation, 3x + 5y = 1, we can rearrange it to solve for x: 3x = 1 – 5y x = (1 – 5y) / 3
Now that we have an expression for x in terms of y, we can substitute it into our second equation: 7((1 – 5y) / 3) + 4y = -13
By simplifying and solving this equation step by step, we can determine the value of y that satisfies both equations and gives us a solution to the system.
As you work through these calculations, remember to carefully distribute any coefficients and combine like terms. Solving systems of equations requires attention to detail but can ultimately lead us to uncovering valuable insights about mathematical relationships.
With perseverance and careful manipulation of equations, you’ll arrive at a numerical value for y that corresponds to a solution satisfying both equations. Don’t hesitate to double-check your work if needed and consider seeking assistance from tutors or online resources if you encounter any difficulties along the way.
In summary, solving systems of equations involves various techniques such as substitution. By isolating one variable and substituting it into another equation, we can find solutions that satisfy both equations simultaneously. Keep practicing your problem-solving skills, and soon you’ll be tackling more complex systems with confidence.