QuestionLet x represent one number and let y represent the other number. Four times a first number decreased by a second number is -13 . The first number increased by twice the second number is 17 . Use the given conditions to write a system of equations. Solve the system and find the numbers.
Studdy Solution
STEP 1
1. Let represent the first number.
2. Let represent the second number.
3. Four times the first number decreased by the second number is -13.
4. The first number increased by twice the second number is 17.
5. We need to write a system of equations based on these conditions and solve for and .
STEP 2
1. Translate the word problem into mathematical equations.
2. Write the system of equations.
3. Solve the system of equations using substitution or elimination.
4. Find the values of and .
STEP 3
Translate the word problem into mathematical equations.
For the first condition: "Four times a first number decreased by a second number is -13", we write:
For the second condition: "The first number increased by twice the second number is 17", we write:
STEP 4
Write the system of equations.
The system of equations is:
\begin{align*}
1) & \quad 4x - y = -13 \\
2) & \quad x + 2y = 17
\end{align*}
STEP 5
Solve the system of equations using substitution or elimination. Here, we will use the elimination method.
First, multiply equation (2) by 2 to align the coefficients of :
Now, we have:
\begin{align*}
1) & \quad 4x - y = -13 \\
3) & \quad 2x + 4y = 34
\end{align*}
STEP 6
Continue solving by eliminating .
Multiply equation (1) by 2 to align the coefficients of :
Now, subtract equation (3) from the new equation:
\begin{align*}
8x - 2y &= -26 \\
-(2x + 4y &= 34)
\end{align*}
This gives:
Divide the entire equation by 6:
STEP 7
Substitute into equation (2) to solve for .
Substitute into equation (2):
Add 10 to both sides:
Divide by 3:
STEP 8
Now substitute back into the expression for .
Substitute into :
The numbers are:
and
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