Solve the System of Equations: -5x+9y=18 and x+8y=16 Using Substitution

Linear Equations with Substitution Method

Find the value of x and and band the substitution method.

{5x+9y=18x+8y=16 \begin{cases} -5x+9y=18 \\ x+8y=16 \end{cases}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve the system of equations
00:04 Isolate X
00:12 This is the expression for X, substitute in the second equation to find Y
00:33 Properly expand brackets, multiply by each factor
00:46 Isolate Y
00:58 Collect like terms
01:17 This is the value of Y
01:22 Now substitute the value of Y to find X
01:40 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find the value of x and and band the substitution method.

{5x+9y=18x+8y=16 \begin{cases} -5x+9y=18 \\ x+8y=16 \end{cases}

2

Step-by-step solution

To solve the given system of linear equations using the substitution method, follow these steps:

  • Step 1: Solve the second equation for x x .

From the second equation:

x+8y=16 x + 8y = 16

We can solve for x x as follows:

x=168y x = 16 - 8y
  • Step 2: Substitute the expression for x x into the first equation.

Substitute x=168y x = 16 - 8y into the first equation:

5(168y)+9y=18 -5(16 - 8y) + 9y = 18

Simplify and solve for y y :

- Distribute 5-5:

80+40y+9y=18 -80 + 40y + 9y = 18

- Combine like terms:

49y80=18 49y - 80 = 18

- Add 80 to both sides:

49y=98 49y = 98

- Divide by 49:

y=9849=2 y = \frac{98}{49} = 2
  • Step 3: Substitute y=2 y = 2 back into the expression for x x .

The expression for x x is:

x=168y x = 16 - 8y

- Substitute y=2 y = 2 :

x=168(2) x = 16 - 8(2) x=1616 x = 16 - 16 x=0 x = 0

Therefore, the values that satisfy both equations in the system are x=0 x = 0 and y=2 y = 2 .

3

Final Answer

x=0,y=2 x=0,y=2

Key Points to Remember

Essential concepts to master this topic
  • Method: Solve one equation for a variable, then substitute
  • Technique: From x + 8y = 16, get x = 16 - 8y
  • Check: Substitute x = 0, y = 2: -5(0) + 9(2) = 18 ✓

Common Mistakes

Avoid these frequent errors
  • Substituting into the same equation you solved
    Don't substitute x = 16 - 8y back into x + 8y = 16 = you get 16 = 16 always! This tells you nothing. Always substitute into the OTHER equation to find the unknown variable.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equations:

\( \begin{cases} 2x+y=9 \\ x=5 \end{cases} \)

FAQ

Everything you need to know about this question

Which equation should I solve for a variable first?

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Choose the equation where a variable has a coefficient of 1 (like x in x + 8y = 16). This avoids fractions and makes the algebra much simpler!

What if I get confused with all the negative signs?

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Take it one step at a time! When distributing -5(16 - 8y), write it as: -5 × 16 + (-5) × (-8y) = -80 + 40y. This helps avoid sign errors.

How do I know if my solution works for both equations?

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Always test your answer in both original equations. For x = 0, y = 2: Check -5(0) + 9(2) = 18 ✓ and 0 + 8(2) = 16 ✓

Can I solve for y first instead of x?

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Yes! But it's usually harder because you'd get y=16x8 y = \frac{16-x}{8} from the second equation, creating fractions. Always pick the easier variable to isolate first.

What if I get a different answer when I check?

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Go back and check your algebra! Common errors include: wrong signs when distributing, not combining like terms correctly, or arithmetic mistakes when dividing.

Is there a faster way than substitution?

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Yes! You could use elimination method by multiplying equations to eliminate variables. But substitution is often clearer when one variable has a coefficient of 1.

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