Question

What is the solution to this system of equations?

One-fourth x + 1 and one-half y = StartFraction 5 Over 8 EndFraction. Three-fourths x minus 1 and one-half y = 3 and StartFraction 3 Over 8 EndFraction

Answer 1

To solve this system of equations, we can use the method of elimination. Multiply the first equation by 2 and the second equation by 1 to make the coefficients of y the same. Then, subtract the second equation from the first to eliminate y. Finally, solve for x and y to find the solution.

Step-by-step explanation:

To solve this system of equations, we can use the method of elimination. Let's eliminate the variable y. Multiply the first equation by 2 and the second equation by 1 to make the coefficients of y the same:

2×(\frac{1}{4}x + 1\frac{1}{2}y) = 2×\frac{5}{8}

\frac{1}{2}x + 1y = \frac{5}{4}

1×(\frac{3}{4}x - 1\frac{1}{2}y) = 1×\frac{3}{8}

\frac{3}{4}x - 1y = \frac{3}{8}

Now, subtract the second equation from the first to eliminate y:

(\frac{1}{2}x + 1y) - (\frac{3}{4}x - 1y) = (\frac{5}{4}) - (\frac{3}{8})

-\frac{1}{4}x + 2y = \frac{7}{8}

Next, multiply the first equation by 8 to eliminate the fractions:

8×(\frac{1}{2}x + 1y) = 8×\frac{5}{4}

4x + 8y = 10

Now we have a system of two equations:

-\frac{1}{4}x + 2y = \frac{7}{8}

4x + 8y = 10

From the first equation, isolate x:

-\frac{1}{4}x = \frac{7}{8} - 2y

Multiply both sides by -4 to get rid of the fraction:

x = -\frac{7}{2} + 8y

Now substitute this value of x into the second equation:

4(-\frac{7}{2} + 8y) + 8y = 10

-28 + 32y + 8y = 10

Combine like terms:

40y = 38

Divide both sides by 40:

y = \frac{19}{20}

Now substitute this value of y back into the first equation to find x:

x = -\frac{7}{2} + 8(\frac{19}{20})

x = -\frac{7}{2} + \frac{152}{20}

Combine like terms:

x = \frac{61}{20}

So the solution to the system of equations is x = \frac{61}{20} and y = \frac{19}{20}.

Answer 2

The solution to the given system of equations is x = 4 and y = -3/4, obtained by clearing the fractions, eliminating one variable, and substituting back to find the other.

Step-by-step explanation:

To solve the given system of equations, we can use the method of substitution or elimination. Let's denote the first equation as Equation (1) and the second one as Equation (2):

1. (1/4)x + (1/2)y = 5/8
2. (3/4)x - (1/2)y = 3 3/8

First, let's clear the fractions by multiplying each term in both equations by 8, the least common multiple of the denominators:

1. 2x + 4y = 5 (multiplied by 8)
2. 6x - 4y = 27 (multiplied by 8)

To eliminate one variable, we can add the two resulting equations:

1. (2x + 4y) + (6x - 4y) = 5 + 27
2. 8x = 32
3. x = 4 (dividing both sides by 8)

Having found the value of x, we substitute it back into either original equation to find y. We can use Equation (1):

(1/4)*4 + (1/2)y = 5/8

1 + (1/2)y = 5/8

(1/2)y = 5/8 - 8/8

(1/2)y = -3/8

Now, multiply both sides by 2 to solve for y:

y = -3/4

The solution to the system of equations is x = 4 and y = -3/4.