What is the solution and steps to 3x+5y=25 𝑦

Let's solve the system of equations using substitution.

The given equations are:

1. $3x + 5y = 25$
2. $y = \frac{1}{2}x + 3$

Step-by-Step Solution

Step 1: Substitute the expression for $y$ from the second equation into the first equation.

Since $y = \frac{1}{2}x + 3$, we replace $y$ in the equation $3x + 5y = 25$.

$3x + 5\left(\frac{1}{2}x + 3\right) = 25$

Step 2: Distribute the 5 to both terms inside the parentheses.

$3x + \frac{5}{2}x + 15 = 25$

Step 3: Combine like terms involving $x$.

$\left(3 + \frac{5}{2}\right)x + 15 = 25$

We rewrite $3$ as $\frac{6}{2}$, so:

$\left(\frac{6}{2} + \frac{5}{2}\right)x + 15 = 25$

$\frac{11}{2}x + 15 = 25$

Step 4: Isolate the $x$-term.

$\frac{11}{2}x = 25 - 15$ $\frac{11}{2}x = 10$

Step 5: Solve for $x$.

Multiply both sides by 2 to get rid of the fraction:

$11x = 20$

Now, divide by 11:

$x = \frac{20}{11}$

Step 6: Substitute the value of $x$ into the equation $y = \frac{1}{2}x + 3$.

$y = \frac{1}{2}\left(\frac{20}{11}\right) + 3$

Simplify:

$y = \frac{20}{22} + 3 = \frac{10}{11} + 3$

Convert 3 to a fraction with denominator 11:

$y = \frac{10}{11} + \frac{33}{11} = \frac{43}{11}$

Final Answer

The solution to the system of equations is:

$x = \frac{20}{11}, \quad y = \frac{43}{11}$

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Would you like further clarification on any step?

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Here are five related questions you might find useful:

1. How do you solve systems of equations by elimination?
2. What are the conditions for a system of equations to have no solution?
3. How do you graph linear equations to find solutions visually?
4. How do you interpret the intersection of two lines algebraically and geometrically?
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Tip: Always double-check your algebra when substituting fractions to avoid mistakes.