Solve the System of Equations: -8x + 3y = 7 and 24x + y = 3

Q: Why multiply by 3 instead of some other number?

We multiply by 3 to make the y coefficients the same (both become 3y). This lets us eliminate y by subtraction. Choose multipliers that create matching coefficients for the variable you want to eliminate.

Q: How do I know which variable to eliminate first?

Look for the easiest path! In this problem, multiplying the second equation by 3 gives us matching y coefficients. Always choose the elimination that requires the simplest multiplication.

Q: Can I use substitution instead of elimination?

Yes! You could solve $y = 3 - 24x$ from equation 2, then substitute into equation 1. Both methods work, but elimination often involves easier arithmetic when coefficients align well.

Q: How do I check if my solution is correct?

Substitute both values into both original equations. For $x = 0.025, y = 2.4$:Equation 1: $-8(0.025) + 3(2.4) = -0.2 + 7.2 = 7$ ✓Equation 2: $24(0.025) + 2.4 = 0.6 + 2.4 = 3$ ✓

Q: Why did the explanation have errors at first?

The initial approach had sign errors during elimination. This shows why it's crucial to work carefully with positive and negative terms. When subtracting equations, distribute the negative sign to every term in the second equation.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the system of equations

00:03 Multiply by 3 so that by subtracting we can isolate Y

00:14 Now this is the system of equations

00:22 Subtract between the equations

00:34 Collect terms

00:43 Isolate Y

00:53 This is the value of Y

01:02 Now let's substitute Y's value to find X

01:12 Isolate 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

Solve the above set of equations and choose the correct answer.

$\begin{cases} -8x+3y=7 \\ 24x+y=3 \end{cases}$

2

Step-by-step solution

We will solve the system of equations using the elimination method.

Step 1: We have the system of equations:

Equation 1: $-8x + 3y = 7$
Equation 2: $24x + y = 3$

Step 2: Let's eliminate $x$ by aligning coefficients. Multiply Equation 1 by 3:

Equation 1: $-8x + 3y = 7$ becomes $-24x + 9y = 21$

Now subtract Equation 2 from the modified Equation 1:

$-24x + 9y - (24x + y) = 21 - 3$

Simplifying, we get:

$-48x + 8y = 18$

Notice, this was incorrect since subtraction led to an error in understanding coefficients. Let's find $y$ directly.

We have:

Equation 1: $-8x + 3y = 7$
Equation 2: $24x + y = 3$

Step 3: Solve for $y$ from Equation 2:

Multiply Equation 2 by 3:

$24x + y = 3$

3 $(24x + y = 3)$ gives:

$72x + 3y = 9$

Subtracting Equation 1 from this new Equation gives:

$(72x + 3y) - (-8x + 3y) = 9 - 7$

$80x = 2$

Step 4: Solve for $x$ :

$x = \frac{2}{80} = 0.025$

Step 5: Substitute $x = 0.025$ back into Equation 2 to find $y$ :

$24(0.025) + y = 3$

$0.6 + y = 3$

$y = 3 - 0.6 = 2.4$

Thus, the solution to the system of equations is $x = 0.025$ and $y = 2.4$ .

The choice corresponding to this solution is:

$x = 0.025, y = 2.4$

3

Final Answer

$x=0.025,y=2.4$

Key Points to Remember

Essential concepts to master this topic

Elimination Rule: Make coefficients of one variable opposites, then add equations
Technique: Multiply equation by 3: $24x + y = 3$ becomes $72x + 3y = 9$
Check: Substitute both values back: $-8(0.025) + 3(2.4) = 7$ ✓

Common Mistakes

Avoid these frequent errors

Making algebraic errors when multiplying equations
Don't multiply incorrectly or forget to distribute to all terms = wrong coefficients! This creates false equations that don't match the original system. Always multiply every term carefully and double-check your arithmetic before eliminating variables.

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 multiply by 3 instead of some other number?

+

We multiply by 3 to make the y coefficients the same (both become 3y). This lets us eliminate y by subtraction. Choose multipliers that create matching coefficients for the variable you want to eliminate.

How do I know which variable to eliminate first?

+

Look for the easiest path! In this problem, multiplying the second equation by 3 gives us matching y coefficients. Always choose the elimination that requires the simplest multiplication.

What if I get decimal answers instead of whole numbers?

+

Decimal solutions are completely normal! Many real-world problems have non-integer answers. Just make sure to calculate carefully and verify your solution by substituting back.

Can I use substitution instead of elimination?

+

Yes! You could solve $y = 3 - 24x$ from equation 2, then substitute into equation 1. Both methods work, but elimination often involves easier arithmetic when coefficients align well.

How do I check if my solution is correct?

+

Substitute both values into both original equations. For $x = 0.025, y = 2.4$ :

Equation 1: $-8(0.025) + 3(2.4) = -0.2 + 7.2 = 7$ ✓
Equation 2: $24(0.025) + 2.4 = 0.6 + 2.4 = 3$ ✓

Why did the explanation have errors at first?

+

The initial approach had sign errors during elimination. This shows why it's crucial to work carefully with positive and negative terms. When subtracting equations, distribute the negative sign to every term in the second equation.

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Solve the System of Equations: -8x + 3y = 7 and 24x + y = 3

System of Equations with Elimination Method

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