Solve the System of Equations: 3x - y = -10 and 9x - 3y = -15

Q: How can I tell if a system has no solution before solving?

Look for proportional coefficients with different constants! If one equation is a multiple of another but the constants don't match, you have parallel lines that never intersect.

Q: What's the difference between no solution and infinite solutions?

No solution: Parallel lines (same slope, different intercepts). Infinite solutions: Same line written differently (all coefficients proportional including constants).

Q: Can I use substitution method instead?

Yes! From the first equation: $y = 3x + 10$. Substitute into the second equation and you'll get $-30 = -15$, which is impossible.

Q: What does it mean geometrically when there's no solution?

The two equations represent parallel lines that never intersect. Since a solution is an intersection point, parallel lines have no solution!

Inconsistent Linear Systems with Elimination

Solve the following system of equations:

$\begin{cases} 3x-y=-10 \\ 9x-3y=-15 \end{cases}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:04 Let's multiply one of the equations by 3, so we can subtract between them

00:22 Now let's subtract between the equations

00:37 Let's group like terms

00:46 We got an illogical expression, therefore there is no solution

00:52 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following system of equations:

$\begin{cases} 3x-y=-10 \\ 9x-3y=-15 \end{cases}$

Step-by-step solution

To solve this problem, let's apply the elimination method:

Step 1: Write down the given equations: $3x - y = -10$ and $9x - 3y = -15$ .
Step 2: Observe that the second equation is 3 times the first equation. Multiply the first equation by 3 for alignment: $3(3x - y) = 3(-10)$ becomes $9x - 3y = -30$ .
Step 3: Compare the two resulting equations: $9x - 3y = -30$ and $9x - 3y = -15$ .
Step 4: Notice these equations suggest a contradiction as the left-hand side is the same ( $9x - 3y$ ), but the right-hand sides are different ( $-30$ vs. $-15$ ).

As the comparison shows a contradiction, these lines are parallel and do not intersect.
Therefore, the system of equations has no solution.

Final Answer

There is no solution.

Key Points to Remember

Essential concepts to master this topic

Parallel Lines Rule: Same slopes, different intercepts mean no solution
Elimination Technique: Multiply first equation by 3: $9x - 3y = -30$
Contradiction Check: Compare $9x - 3y = -30$ vs $9x - 3y = -15$ ✓

Common Mistakes

Avoid these frequent errors

Assuming equations always have solutions
Don't automatically solve for x and y without checking for contradictions = wrong conclusion! When left sides are identical but right sides differ, no solution exists. Always compare the simplified equations first to identify contradictions.

Practice Quiz

Test your knowledge with interactive questions

$\begin{cases} x+y=8 \\ x-y=6 \end{cases}$

FAQ

Everything you need to know about this question

How can I tell if a system has no solution before solving?

Look for proportional coefficients with different constants! If one equation is a multiple of another but the constants don't match, you have parallel lines that never intersect.

What's the difference between no solution and infinite solutions?

No solution: Parallel lines (same slope, different intercepts). Infinite solutions: Same line written differently (all coefficients proportional including constants).

Why did multiplying the first equation by 3 help?

Multiplying by 3 made the coefficients identical to the second equation. This lets you easily compare whether the equations represent the same line or parallel lines!

Can I use substitution method instead?

Yes! From the first equation: $y = 3x + 10$ . Substitute into the second equation and you'll get $-30 = -15$ , which is impossible.

What does it mean geometrically when there's no solution?

The two equations represent parallel lines that never intersect. Since a solution is an intersection point, parallel lines have no solution!

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