Solve This System of Equations: Find x and y in x - y = 5 and 2x - 3y = 8

Q: Why multiply the first equation by 2 instead of something else?

We multiply by 2 to make the x-coefficients equal in both equations. Since the first equation has $x$ and the second has $2x$, multiplying by 2 gives us $2x$ in both equations.

Q: Can I eliminate y instead of x?

Absolutely! You could multiply the first equation by -3 to get $-3x + 3y = -15$, then add it to the second equation. Both methods work - choose whichever looks easier!

Q: How do I know which method is best for solving systems?

Elimination: Good when coefficients can be easily made equalSubstitution: Good when one equation is already solved for a variableGraphing: Helpful for visualizing, but less precise

System of Equations with Elimination Method

Solve the following system of equations:

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

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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

00:05 Multiply the first equation by 2

00:18 Subtract between the equations

00:34 Negative times negative always equals positive

00:40 Collect like terms

00:49 This is the solution for Y

00:54 Isolate X from the first equation

01:02 Substitute the Y value we found and solve for X

01:11 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

Solve the following system of equations:

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

Step-by-step solution

To solve this system of linear equations using the elimination method, we will follow these steps:

Step 1: Align the equations for elimination.

Write the equations as they are given:

$x - y = 5$ (Equation 1)

$2x - 3y = 8$ (Equation 2)

Step 2: Eliminate one variable.

Multiply Equation 1 by 2 to align the coefficient of $x$ with that in Equation 2:

$2(x - y) = 2 \times 5$

Thus, the transformed Equation 1 is:

$2x - 2y = 10$ (Equation 3)

Subtract Equation 2 from Equation 3 to eliminate $x$ :

$(2x - 2y) - (2x - 3y) = 10 - 8$

This simplifies to:

$y = 2$

Step 3: Solve for the other variable.

Substitute $y = 2$ into Equation 1 to solve for $x$ .

$x - 2 = 5$

Solve for $x$ by adding 2 to both sides:

$x = 7$

Therefore, the solution to the system of linear equations is $\mathbf{x = 7}$ and $\mathbf{y = 2}$ .

This solution matches the choice:

$x = 7, y = 2$

Final Answer

$x=7,y=2$

Key Points to Remember

Essential concepts to master this topic

Setup: Align equations and choose which variable to eliminate first
Technique: Multiply first equation by 2: $2x - 2y = 10$
Check: Substitute $x = 7, y = 2$ into both original equations ✓

Common Mistakes

Avoid these frequent errors

Subtracting equations incorrectly
Don't subtract $(2x - 3y) - (2x - 2y)$ = wrong variable eliminated! This gives you $-y$ instead of $y$ and leads to incorrect answers. Always subtract the second equation from the first: $(2x - 2y) - (2x - 3y)$ .

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 the first equation by 2 instead of something else?

We multiply by 2 to make the x-coefficients equal in both equations. Since the first equation has $x$ and the second has $2x$ , multiplying by 2 gives us $2x$ in both equations.

Can I eliminate y instead of x?

Absolutely! You could multiply the first equation by -3 to get $-3x + 3y = -15$ , then add it to the second equation. Both methods work - choose whichever looks easier!

What if I get different answers when I check my solution?

If your solution doesn't work in both original equations, you made an arithmetic error. Go back and carefully check each step, especially when multiplying equations and combining like terms.

Do I always need to multiply equations in elimination?

Not always! Sometimes the coefficients already line up nicely for elimination. But in this problem, we need to multiply to make the coefficients of one variable the same.

How do I know which method is best for solving systems?

Elimination: Good when coefficients can be easily made equal
Substitution: Good when one equation is already solved for a variable
Graphing: Helpful for visualizing, but less precise

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