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Answer the questions and complete the steps to solve the system of equations below.

3x + 2y = 16
y=x+3
a) Which variable should you substitute for in the equation 3x + 2y = 16?
choose your answer...
b) What value can be substituted for that variable? choose your answer...
c) Rewrite the equation with the substitution. choose your answer...
choose your answer...
3(x+3) + 2y =16
• x into the equation and solve for y.
3x + 2(x+3) = 16

3 answers

To solve the system of equations, let's go through the steps one by one.

  1. Identify which variable to substitute:

    The second equation is \(y = x + 3\). It is convenient to substitute for \(y\) in the first equation because it directly gives \(y\) in terms of \(x\).

    a) Which variable should you substitute for in the equation \(3x + 2y = 16\)?
    Answer: \(y\)

  2. What value can be substituted for that variable?

    From the second equation, we have \(y = x + 3\).

    b) What value can be substituted for that variable?
    Answer: \(x + 3\)

  3. Rewrite the equation with the substitution:

    Replace \(y\) in the first equation with \(x + 3\).

    c) Rewrite the equation with the substitution.

    Start with the original equation: \[ 3x + 2y = 16 \] Substitute \(y\) with \(x + 3\): \[ 3x + 2(x + 3) = 16 \]

  4. Now, simplify and solve for \(x\):

    \[ 3x + 2(x + 3) = 16 \] Expanding this: \[ 3x + 2x + 6 = 16 \] Combine like terms: \[ 5x + 6 = 16 \]

    Next, isolate \(x\) by subtracting 6 from both sides: \[ 5x = 10 \]

    Finally, divide by 5: \[ x = 2 \]

  5. Substitute \(x = 2\) back into the \(y\) equation to find \(y\):

    \[ y = x + 3 = 2 + 3 = 5 \]

Thus, the solution to the system of equations is: \[ (x, y) = (2, 5) \]

Answer the questions and complete the steps to solve the system of equations below.
3x + 2y = 16
y =x+3
a) Which variable should you substitute for in the equation 3x + 2y = 16?
choose your answer...
b) What value can be substituted for that variable? choose your answer... V
c) Rewrite the equation with the substitution. choose your answer...
d) Solve the equation for x.
e) Substitute that value for x into the equation and solve for y.
f) Use substitution to check your solution for the system.

Let’s go through the process step-by-step based on the given system of equations:

  • Equation 1: \(3x + 2y = 16\)
  • Equation 2: \(y = x + 3\)

a) Which variable should you substitute for in the equation \(3x + 2y = 16\)?

Since Equation 2 expresses \(y\) in terms of \(x\), it is most convenient to substitute for \(y\) in Equation 1.

Answer: \(y\)

b) What value can be substituted for that variable?

From Equation 2, \(y\) can be replaced with \(x + 3\).

Answer: \(x + 3\)

c) Rewrite the equation with the substitution.

Now, substitute \(y\) in Equation 1 with \(x + 3\):

Starting with: \[ 3x + 2y = 16 \] Substituting \(y\): \[ 3x + 2(x + 3) = 16 \]

d) Solve the equation for \(x\).

Now, we will simplify and solve for \(x\): \[ 3x + 2(x + 3) = 16 \] Expanding this: \[ 3x + 2x + 6 = 16 \] Combine like terms: \[ 5x + 6 = 16 \] Now, isolate \(x\) by subtracting 6 from both sides: \[ 5x = 10 \] Finally, divide by 5: \[ x = 2 \]

e) Substitute that value for \(x\) into the equation and solve for \(y\).

Now substitute \(x = 2\) back into Equation 2 to find \(y\): \[ y = x + 3 = 2 + 3 = 5 \]

f) Use substitution to check your solution for the system.

Now we need to check if our solution \((2, 5)\) satisfies both original equations.

Checking in Equation 1: \[ 3x + 2y = 16 \] Substituting \(x = 2\) and \(y = 5\): \[ 3(2) + 2(5) = 6 + 10 = 16 \quad \text{(True)} \]

Checking in Equation 2: \[ y = x + 3 \] Substituting \(x = 2\): \[ 5 = 2 + 3 \quad \text{(True)} \]

Since both equations are satisfied, our solution \((x, y) = (2, 5)\) is correct.

Final Summary:

  • a: \(y\)
  • b: \(x + 3\)
  • c: \(3x + 2(x + 3) = 16\)
  • d: \(x = 2\)
  • e: \(y = 5\)
  • f: Both equations are satisfied, verifying the solution \((2, 5)\).