Exercise: Find the Ratio x : y

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for finding the ratio $x:y$ for each problem.

1. $x^2 - 4xy + 3y^2 = 0$

Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).
$\frac{x^2}{y^2} - \frac{4xy}{y^2} + \frac{3y^2}{y^2} = 0$ $\left(\frac{x}{y}\right)^2 - 4\left(\frac{x}{y}\right) + 3 = 0$
Let $k = \frac{x}{y}$ .
Substitute $k$ into the equation and solve the quadratic.
$k^2 - 4k + 3 = 0$ $(k-1)(k-3) = 0$
Solve for $k$ .
$k-1=0 \implies k=1$ $k-3=0 \implies k=3$
Substitute back $k = \frac{x}{y}$ and express as a ratio.
$\frac{x}{y} = 1 \quad \text{or} \quad \frac{x}{y} = 3$
The ratio $x:y$ is $\boxed{\text{1:1 or 3:1}}$ .

2. $2x^2 - 5xy + 2y^2 = 0$
Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).
$2\left(\frac{x}{y}\right)^2 - 5\left(\frac{x}{y}\right) + 2 = 0$
Let $k = \frac{x}{y}$ .
Substitute $k$ into the equation and solve the quadratic.
$2k^2 - 5k + 2 = 0$ $(2k-1)(k-2) = 0$
Solve for $k$ .
$2k-1=0 \implies k=\frac{1}{2}$ $k-2=0 \implies k=2$
Substitute back $k = \frac{x}{y}$ and express as a ratio.
$\frac{x}{y} = \frac{1}{2} \quad \text{or} \quad \frac{x}{y} = 2$
The ratio $x:y$ is $\boxed{\text{1:2 or 2:1}}$ .

3. $3x^2 + xy - 2y^2 = 0$
Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).
$3\left(\frac{x}{y}\right)^2 + \left(\frac{x}{y}\right) - 2 = 0$
Let $k = \frac{x}{y}$ .
Substitute $k$ into the equation and solve the quadratic.
$3k^2 + k - 2 = 0$ $(3k-2)(k+1) = 0$
Solve for $k$ .
$3k-2=0 \implies k=\frac{2}{3}$ $k+1=0 \implies k=-1$
Substitute back $k = \frac{x}{y}$ and express as a ratio.
$\frac{x}{y} = \frac{2}{3} \quad \text{or} \quad \frac{x}{y} = -1$
The ratio $x:y$ is $\boxed{\text{2:3 or -1:1}}$ .

4. $\frac{x+3y}{x} = \frac{7}{4}$
Separate the fraction on the left side (assuming $x \neq 0$ ).
$\frac{x}{x} + \frac{3y}{x} = \frac{7}{4}$ $1 + 3\frac{y}{x} = \frac{7}{4}$
Isolate $\frac{y}{x}$ .
$3\frac{y}{x} = \frac{7}{4} - 1$ $3\frac{y}{x} = \frac{7}{4} - \frac{4}{4}$ $3\frac{y}{x} = \frac{3}{4}$
Solve for $\frac{y}{x}$ and then find $\frac{x}{y}$ .
$\frac{y}{x} = \frac{1}{4}$ $\frac{x}{y} = 4$
Express as a ratio.
The ratio $x:y$ is $\boxed{\text{4:1}}$ .

5. $\frac{2x-y}{y} = 5$
Separate the fraction on the left side (assuming $y \neq 0$ ).
$\frac{2x}{y} - \frac{y}{y} = 5$ $2\frac{x}{y} - 1 = 5$
Isolate $\frac{x}{y}$ .
$2\frac{x}{y} = 5 + 1$ $2\frac{x}{y} = 6$
Solve for $\frac{x}{y}$ .
$\frac{x}{y} = 3$
Express as a ratio.
The ratio $x:y$ is $\boxed{\text{3:1}}$ .

6. $\frac{3x}{y} + \frac{8y}{x} = 10$
Let $k = \frac{x}{y}$ .
Then $\frac{y}{x} = \frac{1}{k}$ . Substitute these into the equation (assuming $x, y \neq 0$ ).
$3k + \frac{8}{k} = 10$
Multiply the entire equation by $k$ to eliminate the denominator.
$3k^2 + 8 = 10k$
Rearrange into a quadratic equation and solve for $k$ .
$3k^2 - 10k + 8 = 0$ $(3k-4)(k-2) = 0$
Solve for $k$ .
$3k-4=0 \implies k=\frac{4}{3}$ $k-2=0 \implies k=2$
Substitute back $k = \frac{x}{y}$ and express as a ratio.
$\frac{x}{y} = \frac{4}{3} \quad \text{or} \quad \frac{x}{y} = 2$
The ratio $x:y$ is $\boxed{\text{4:3 or 2:1}}$ .

7. $4x^2 - 12xy + 9y^2 = 0$
Recognize the equation as a perfect square trinomial.
$(2x)^2 - 2(2x)(3y) + (3y)^2 = 0$ $(2x - 3y)^2 = 0$
Take the square root of both sides.
$2x - 3y = 0$
Isolate $\frac{x}{y}$ .
$2x = 3y$ $\frac{x}{y} = \frac{3}{2}$
Express as a ratio.
The ratio $x:y$ is $\boxed{\text{3:2}}$ .

8. $\frac{5x+y}{2x} = 2$
Multiply both sides by $2x$ (assuming $x \neq 0$ ).
$5x+y = 2(2x)$ $5x+y = 4x$
Isolate $x$ and $y$ terms.
$y = 4x - 5x$ $y = -x$
Solve for $\frac{x}{y}$ .
$\frac{x}{y} = -1$
Express as a ratio.
The ratio $x:y$ is $\boxed{\text{-1:1}}$ .

9. $\frac{x}{2y} + \frac{12y}{x} = 7$
Let $k = \frac{x}{y}$ .
Then $\frac{x}{2y} = \frac{1}{2}k$ and $\frac{12y}{x} = \frac{12}{k}$ . Substitute these into the equation (assuming $x, y \neq 0$ ).
$\frac{1}{2}k + \frac{12}{k} = 7$
Multiply the entire equation by $2k$ to eliminate the denominators.
$k^2 + 24 = 14k$
Rearrange into a quadratic equation and solve for $k$ .
$k^2 - 14k + 24 = 0$ $(k-2)(k-12) = 0$
Solve for $k$ .
$k-2=0 \implies k=2$ $k-12=0 \implies k=12$
Substitute back $k = \frac{x}{y}$ and express as a ratio.
$\frac{x}{y} = 2 \quad \text{or} \quad \frac{x}{y} = 12$
The ratio $x:y$ is $\boxed{\text{2:1 or 12:1}}$ .

10. $5x^2 - 13xy + 6y^2 = 0$
Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).
$5\left(\frac{x}{y}\right)^2 - 13\left(\frac{x}{y}\right) + 6 = 0$
Let $k = \frac{x}{y}$ .
Substitute $k$ into the equation and solve the quadratic.
$5k^2 - 13k + 6 = 0$ $(5k-3)(k-2) = 0$
Solve for $k$ .
$5k-3=0 \implies k=\frac{3}{5}$ $k-2=0 \implies k=2$
Substitute back $k = \frac{x}{y}$ and express as a ratio.
$\frac{x}{y} = \frac{3}{5} \quad \text{or} \quad \frac{x}{y} = 2$
The ratio $x:y$ is $\boxed{\text{3:5 or 2:1}}$ .

Exercise: Find the Ratio x : y

Step-by-step solution

Divide the entire equation by y2y^2y2 (assuming y≠0y \neq 0y=0).

Let k=xyk = \frac{x}{y}k=yx​.

Solve for kkk.

Substitute back k=xyk = \frac{x}{y}k=yx​ and express as a ratio.

Divide the entire equation by y2y^2y2 (assuming y≠0y \neq 0y=0).

Let k=xyk = \frac{x}{y}k=yx​.

Solve for kkk.

Substitute back k=xyk = \frac{x}{y}k=yx​ and express as a ratio.

Divide the entire equation by y2y^2y2 (assuming y≠0y \neq 0y=0).

Let k=xyk = \frac{x}{y}k=yx​.

Solve for kkk.

Substitute back k=xyk = \frac{x}{y}k=yx​ and express as a ratio.

Separate the fraction on the left side (assuming x≠0x \neq 0x=0).

Isolate yx\frac{y}{x}xy​.

Solve for yx\frac{y}{x}xy​ and then find xy\frac{x}{y}yx​.

Express as a ratio.

Separate the fraction on the left side (assuming y≠0y \neq 0y=0).

Isolate xy\frac{x}{y}yx​.

Solve for xy\frac{x}{y}yx​.

Express as a ratio.

Let k=xyk = \frac{x}{y}k=yx​.

Multiply the entire equation by kkk to eliminate the denominator.

Rearrange into a quadratic equation and solve for kkk.

Solve for kkk.

Substitute back k=xyk = \frac{x}{y}k=yx​ and express as a ratio.

Recognize the equation as a perfect square trinomial.

Take the square root of both sides.

Isolate xy\frac{x}{y}yx​.

Express as a ratio.

Multiply both sides by 2x2x2x (assuming x≠0x \neq 0x=0).

Isolate xxx and yyy terms.

Solve for xy\frac{x}{y}yx​.

Express as a ratio.

Let k=xyk = \frac{x}{y}k=yx​.

Multiply the entire equation by $2k$ to eliminate the denominators.

Rearrange into a quadratic equation and solve for kkk.

Solve for kkk.

Substitute back k=xyk = \frac{x}{y}k=yx​ and express as a ratio.

Divide the entire equation by y2y^2y2 (assuming y≠0y \neq 0y=0).

Let k=xyk = \frac{x}{y}k=yx​.

Solve for kkk.

Substitute back k=xyk = \frac{x}{y}k=yx​ and express as a ratio.

Still stuck on this one?

More Mathematics Questions

Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).

Let $k = \frac{x}{y}$ .

Solve for $k$ .

Substitute back $k = \frac{x}{y}$ and express as a ratio.

Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).

Let $k = \frac{x}{y}$ .

Solve for $k$ .

Substitute back $k = \frac{x}{y}$ and express as a ratio.

Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).

Let $k = \frac{x}{y}$ .

Solve for $k$ .

Substitute back $k = \frac{x}{y}$ and express as a ratio.

Separate the fraction on the left side (assuming $x \neq 0$ ).

Isolate $\frac{y}{x}$ .

Solve for $\frac{y}{x}$ and then find $\frac{x}{y}$ .

Separate the fraction on the left side (assuming $y \neq 0$ ).

Isolate $\frac{x}{y}$ .

Solve for $\frac{x}{y}$ .

Let $k = \frac{x}{y}$ .

Multiply the entire equation by $k$ to eliminate the denominator.

Rearrange into a quadratic equation and solve for $k$ .

Solve for $k$ .

Substitute back $k = \frac{x}{y}$ and express as a ratio.

Isolate $\frac{x}{y}$ .

Multiply both sides by $2x$ (assuming $x \neq 0$ ).

Isolate $x$ and $y$ terms.

Solve for $\frac{x}{y}$ .

Let $k = \frac{x}{y}$ .

Rearrange into a quadratic equation and solve for $k$ .

Solve for $k$ .

Substitute back $k = \frac{x}{y}$ and express as a ratio.

Divide the entire equation by $y^2$ (assuming $y \neq 0$ ).

Let $k = \frac{x}{y}$ .

Solve for $k$ .

Substitute back $k = \frac{x}{y}$ and express as a ratio.