Solve the System: Find x and y in -5x + y = 8 and 3x - 2y = 11 Using Substitution

Q: How do I handle the negative signs when substituting?

Be extra careful with the distributive property! When you see -2(5x + 8), distribute the -2 to both terms: -2(5x) + (-2)(8) = -10x - 16.

Q: Why are my answers fractions instead of whole numbers?

Not all systems have integer solutions! Fractional answers are completely valid in mathematics. The key is to simplify your fractions and verify they work in both original equations.

Q: Can I use elimination instead of substitution for this problem?

Yes! Both methods work, but substitution is often clearer when one variable has a coefficient of 1 or -1. Elimination might involve more fraction work from the beginning with these particular equations.

Q: How do I check if my fractional answer is correct?

Substitute $ x = -\frac{27}{7} $ and $ y = -\frac{79}{7} $ into both original equations. If both sides equal the same value for each equation, your solution is correct!

System Substitution with Fractional Solutions

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

$\begin{cases} -5x+y=8 \\ 3x-2y=11 \end{cases}$

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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 Y

00:14 This is the expression for Y, substitute in the second equation to find X

00:36 Properly expand brackets, multiply by each factor

00:46 Isolate X

00:53 Collect like terms

01:09 This is the value of X

01:17 Now substitute the value of X to find Y

01:40 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

$\begin{cases} -5x+y=8 \\ 3x-2y=11 \end{cases}$

Step-by-step solution

To solve this system of equations using the substitution method, we follow these steps:

Step 1: Solve for one of the variables in terms of the other using the first equation $-5x + y = 8$ .

We solve for $y$ :

$y = 5x + 8$

Step 2: Substitute the expression for $y$ from Step 1 into the second equation $3x - 2y = 11$ .

$3x - 2(5x + 8) = 11$

Step 3: Simplify and solve for $x$ .

Simplify the substitution:

$3x - 10x - 16 = 11$

$-7x - 16 = 11$

Add 16 to both sides:

$-7x = 27$

Divide by -7:

$x = -\frac{27}{7}$

Step 4: Substitute $x$ back into the expression for $y$ from Step 1.

$y = 5\left(-\frac{27}{7}\right) + 8$

Simplify:

$y = -\frac{135}{7} + \frac{56}{7}$

$y = -\frac{135}{7} + \frac{56}{7} = -\frac{79}{7}$

Therefore, the solution to the system is $x = -\frac{27}{7}$ and $y = -\frac{79}{7}$ .

Final Answer

$x=-\frac{27}{7},y=-\frac{79}{7}$

Key Points to Remember

Essential concepts to master this topic

Substitution Method: Solve one equation for a variable, substitute into other
Technique: From -5x + y = 8, get y = 5x + 8
Check: Substitute $x = -\frac{27}{7}, y = -\frac{79}{7}$ back into both original equations ✓

Common Mistakes

Avoid these frequent errors

Incorrect algebraic manipulation when distributing
Don't substitute y = 5x + 8 and get 3x - 2(5x + 8) = 3x - 10x + 16 = 11! Forgetting the negative sign when distributing gives -16 instead of -16. Always distribute the negative sign: 3x - 2(5x + 8) = 3x - 10x - 16.

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

Why did we solve for y in the first equation instead of x?

We chose y because it has a coefficient of 1 in the first equation, making it easier to isolate. You could solve for x first, but it would involve more fractions from the start.

How do I handle the negative signs when substituting?

Be extra careful with the distributive property! When you see -2(5x + 8), distribute the -2 to both terms: -2(5x) + (-2)(8) = -10x - 16.

Why are my answers fractions instead of whole numbers?

Not all systems have integer solutions! Fractional answers are completely valid in mathematics. The key is to simplify your fractions and verify they work in both original equations.

Can I use elimination instead of substitution for this problem?

Yes! Both methods work, but substitution is often clearer when one variable has a coefficient of 1 or -1. Elimination might involve more fraction work from the beginning with these particular equations.

How do I check if my fractional answer is correct?

Substitute $x = -\frac{27}{7}$ and $y = -\frac{79}{7}$ into both original equations. If both sides equal the same value for each equation, your solution is correct!

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