SAT Math - How to find the solution for a system of equations

If 7_x_ + y = 25 and 6_x_ + y = 23, what is the value of x?

Explanation

You can subtract the second equation from the first equation to eliminate y:

7_x_ + y = 25 – 6_x_ + y = 23: 7_x_ – 6_x_ = x; y – y = 0; 25 – 23 = 2

x = 2

You could also solve one equation for y and substitute that value in for y in the other equation:

6_x_ + y = 23 → y = 23 – 6_x_.

7_x_ + y = 25 → 7_x_ + (23 – 6_x_) = 25 → x + 23 = 25 → x = 2

The sum of four consecutive even integers is , but their product is . What is the least of those integers?

Explanation

Any time the product of consecutive numbers is , must be a one of those consecutive numbers, because if it is not, the product will be non-zero. This leaves us with four possibilities, depending on where is placed in the sequence.

As we can see, , , and are our numbers in question, meaning is our answer as the lowest number.

Note that it is possible to use algebra and set up a system of equations, but it's more time-consuming, which could hinder more than help in a standardized test setting.

How many solutions are there to the following system of equations?

$\pi x + 2\pi y = 3\pi$

$-\pi x -2\pi y = -3\pi$

There are an infinite number of solutions.

There is 1 single solution.

There are 2 solutions.

There are 3 solutions.

There are no solutions.

Explanation

If we use elimination to solve this system of equations, we can add the two equations together. This results in 0=0.

When elimination results in 0=0, that means that the two equations represent the same line. Therefore, there are an infinite number of solutions.

What is the value of in the following system of equations? Round your answer to the hundredths place.

Explanation

You can solve this problem in a number of ways, but one way to solve it is by using substitution. You can begin to do that by solving for in the first equation:

Now, you can substitute in that value of into the second equation and solve for :

Let's consider this equation as adding a negative 3 rather than subtracting a 3 to make distributing easier:

Distribute the negative 3:

We can now combine like variables and solve for :

$x=\frac{41.5}{6.5}=6.384615385\approx6.38$

At what point will the lines $\dpi{100} \small 4x+2=y$ and $\dpi{100} \small 3x+3=y$ intersect?

(1, 6)

(1, –6)

(6, 1)

(–1, 6)

(6, –1)

Explanation

In order to find this point, we must find the solution to the system of equations. we will use substitution, setting the two expressions for y equal to one another.

$\dpi{100} \small 4x+2=3x+3 \rightarrow x=1$

Then we plug this value back into either expression for y, giving us $\dpi{100} \small y=4\times 1+2=6$

So the point is (1, 6).

Solve for the point of intersection of the following two lines:

$\begin{Bmatrix} x-y=2\ 3x+4y=5 \end{Bmatrix}$

$\left(\frac{13}{7},-\frac{1}{7}\right)$

$\left(-\frac{1}{7},\frac{13}{7}\right)$

$\left(\frac{1}{7},\frac{13}{7}\right)$

Explanation

Solve for or first. Let's solve for . To do this, we must eliminate the variables. Multiply the first equation by the coefficient of the variable in the second equation.

$\begin{Bmatrix} 3(x-y=2)\ 3x+4y=5 \end{Bmatrix} = \begin{Bmatrix} 3x-3y=6\ 3x+4y=5 \end{Bmatrix}$

Subtract the second equation from the first equation and solve for .

$y=-\frac{1}{7}$

Resubstitute this value to either original equations. Let's substitute this value into .

$x-(-\frac{1}{7})=2$

$x+\frac{1}{7}=2$

Find the common denominator and solve for the unknown variable.

$\frac{7x}{7}+\frac{1}{7}=\frac{14}{7}$

$x=\frac{13}{7}$

The correct answer is: $(\frac{13}{7},-\frac{1}{7})$

If x2 – y2 = 20, and x + y = 10, then what is the product of x and y?

–64

–4

–24

Explanation

This problem involves a system of two equations. The first equation is x2 – y2 = 20, and the second equation is x + y = 10. Let us solve the second equation in terms of y, and then we can substitute this value into the first equation.

x + y = 10

Subtract y from both sides.

x = 10 – y

Substitute 10 - y for x in the first equation.

x2 – y2 = 20

(10 - y)2 – y2 = 20

We can use the FOIL method to find (10 – y)2.

(10 – y)2 = (10 – y)(10 – y) = 10(10) – 10y – 10y + y2 = 100 –20y + y2.

Now we can go back to our original equation and replace (10 – y)2 with 100 – 20y + y2.

(100 – 20y + y2) – y2 = 20

100 – 20y = 20

Subtract 100 from both sides.

–20y = –80

Divide both sides by –20.

y = 4.

Now that we know that y = 4, we can use either of our original two equations to solve for x. Using the equation x + y = 10 is probably simpler.

x + y = 10

x + 4 = 10

x = 6.

The original question asks for the product of x and y, which would be 4(6), which equals 24.

The answer is 24.

The graph below is the graph of a piece-wise function in some interval. Identify, in interval notation, the decreasing interval.

$\left ( 1,3 \right )$

$\left ( -\infty ,-2\right ]\cup \left ( 1,3 \right )$

$\left ( -\infty ,-2\right )\cup \left ( 1,3\right]$

$(-\infty ,-2]\cup (-\infty ,1)\cup (1,3)$

$\left ( -\infty ,\infty \right )$

Explanation

As is clear from the graph, in the interval between ( included) to , the is constant at and then from ( not included) to ( not included), the is a decreasing function.