x + y = 14; x - y = 4; 3x - 9y = 3; √3x + √3y = 0; 2x + 3y = 11 and 2x - 4y = -24. Solve the set of linear equations.
Understand the Problem
પ્રશ્નમાં કેટલીક રેખીય સમીકરણો આપવામાં આવ્યા છે જે ખોટા છે. અમને આ સમીકરણોને ઠીક કરવાની જરૂર છે અને તે માટેના પગલાં દર્શાવવાની જરૂર છે.
Answer
Corrected equations: 1. $x + y = 14, x - y = 4$ 2. $3x - y = 3$ 3. $x + y = 0$ 4. $y = \frac{3}{2}x + 1$ 5. $2x + 3y = 13$
Answer for screen readers
The corrected equations are as follows:
$$ x + y = 14, \quad x - y = 4 $$ Solution: $x = 9, y = 5$
$$ 3x - y = 3 $$ (valid)
$$ x + y = 0 $$
$$ y = \frac{3}{2}x + 1 $$
$$ 2x + 3y = 13 $$
Steps to Solve
- Adjusting the first equation
Let's analyze the first equation: $$ x + y = 14 $$ $$ x - y = 4 $$
To solve these, we'll add both equations to eliminate $y$: $$ (x + y) + (x - y) = 14 + 4 $$
This simplifies to: $$ 2x = 18 $$
Now solve for $x$: $$ x = 9 $$
Next, substitute $x$ back into one of the original equations to find $y$: $$ 9 + y = 14 \implies y = 14 - 9 $$
Thus, $$ y = 5 $$
- Checking the second equation
For the second equation: $$ 3x - y = 3 $$ $$ 9x - 3y = 9 $$
The second equation can be simplified by dividing by 3: $$ 3x - y = 3 $$
This is the same as the first equation. The second equation is valid, confirming consistency.
- Revising the third equation
In the third equation: $$ \sqrt{3}x + \sqrt{3}y = 0 $$
Factor out $\sqrt{3}$: $$ \sqrt{3}(x + y) = 0 $$
Setting the term in parentheses to zero gives: $$ x + y = 0 $$
- Validating fourth equation
We need to simplify: $$ 3x - 2y = -2 $$
This can be rearranged as: $$ 3x = 2y - 2 \implies y = \frac{3}{2}x + 1 $$
- Analyzing the fifth equation
For the equation: $$ \frac{x}{3} + \frac{y}{2} = \frac{13}{6} $$
Multiply through by 6 to eliminate the denominators: $$ 2x + 3y = 13 $$
That’s the final version of the fifth equation.
The corrected equations are as follows:
$$ x + y = 14, \quad x - y = 4 $$ Solution: $x = 9, y = 5$
$$ 3x - y = 3 $$ (valid)
$$ x + y = 0 $$
$$ y = \frac{3}{2}x + 1 $$
$$ 2x + 3y = 13 $$
More Information
These equations represent a system of linear relationships. Solving them provides the values of $x$ and $y$.
The use of substitution and elimination methods helps to find inconsistent or consistent solutions for multiple equations.
Tips
-
Forgetting to simplify equations: Always check for common factors or ways to simplify complex expressions before proceeding.
-
Incorrect substitutions: Make sure to substitute values back into the correct original equation.
-
Sign errors: Pay careful attention to the signs when manipulating equations.
AI-generated content may contain errors. Please verify critical information
