Question

If \(x+4 y=19\) and \(2 x-y=11,\) what is the value of \(y ?\)

Short answer

The value of \(y\) is 3.

Step-by-step solution

Write Down the Given Equations

The two equations given are:\[x + 4y = 19\] and\[2x - y = 11\]

Express One Variable in Terms of the Other

Solve the second equation for one of the variables. Let's solve for \(y\):\[2x - y = 11\]\[y = 2x - 11\]

Substitute the Expression into the Other Equation

Substitute \(y = 2x - 11\) into the first equation:\[x + 4(2x - 11) = 19\]

Simplify and Solve for \(x\)

Simplify the equation and solve for \(x\):\[x + 8x - 44 = 19\]\[9x - 44 = 19\]\[9x = 63\]\[x = 7\]

Find the Value of \(y\)

Substitute \(x = 7\) back into the expression for \(y\):\[y = 2(7) - 11\]\[y = 14 - 11\]\[y = 3\]

Key concepts

solving equations

Solving equations is a fundamental aspect of algebra. It involves finding the value of an unknown variable that makes the equation true.
In this problem, we are given two linear equations and need to find the value of one variable (y).
To do that, we have to follow a systematic approach, which includes writing down the given equations, expressing one variable in terms of the other, substituting this expression into the other equation, and solving for the unknowns.

substitution method

The substitution method is a technique to solve systems of linear equations.
It involves solving one equation for one variable in terms of the other variable, and then substituting this expression into the second equation.

• First, let's rewrite the given equations: \[x + 4y = 19\] \[2x - y = 11\]

Now, we solve the second equation for y: \[2x - y = 11\] \[y = 2x - 11\]
Next, we substitute this value of y into the first equation and solve for x.

algebraic manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to solve.
Let's perform algebraic manipulations on our equations:

• Start from the substituted equation: \[x + 4(2x - 11) = 19\]

Simplify inside the parentheses: \[x + 8x - 44 = 19\] Combine like terms: \[9x - 44 = 19\] Isolate the variable x: \[9x = 63\] \[x = 7\]
By simplifying and isolating the variable, we can find the solution step by step.
Now that we have the value of x, we can find the corresponding value of y.

educational math problem

Solving educational math problems helps students develop critical thinking and problem-solving skills.
It also allows students to practice various techniques such as substitution and algebraic manipulation, which are crucial for more advanced mathematics.
In this exercise, we solved a system of linear equations to find the value of y.

• First, we determined the value of x: \[x = 7\]
• Next, we found y by substituting x back into the expression for y: \[y = 2(7) - 11\] \[y = 3\]

Through practice and mastery of these steps, students can confidently approach and solve similar algebraic problems.