🧮 algebra

1. **State the problem:** A group of friends is renting a beach house for 9000. They want to split the cost equally, but one friend suggests that each person's share should be equa

1. **State the problem:** We have two linear functions:

1. State the problem: Solve the equation $$-\frac{4}{x + 1} = \frac{5}{3x + 1}$$ for $x$. 2. Cross-multiply to eliminate the denominators:

1. **State the problem:** We have two linear functions: $$y = \frac{1}{2}x - 4$$

1. **State the problem:** Solve the system of linear equations: $$7x + 9y = -2$$

1. **State the problem:** Solve the system of linear equations: $$-7x - 8y = 24$$

1. **State the problem:** Simplify the expression $$\frac{8}{1-2x} + \frac{x}{2x-1}$$. 2. **Identify the denominators:** The denominators are $$1-2x$$ and $$2x-1$$.

1. **State the problem:** Solve the system of linear equations using the elimination method: $$-2x - 5y = -12$$

1. **State the problem:** Solve the system of linear equations using substitution: $$4x + 3y = 26$$

1. **State the problem:** Solve the system of linear equations using substitution: $$3y + 17 = x$$

1. **State the problem:** Simplify the expression $$\frac{1}{x-1} + \frac{-x+1}{x^2}$$. 2. **Rewrite the expression:** The expression is $$\frac{1}{x-1} + \frac{-x+1}{x^2}$$.

1. **State the problem:** Solve the system of linear equations using substitution: $$3x - 5y = 7$$

1. **State the problem:** Solve the system of linear equations: $$x + 4y = 17$$

1. **State the problem:** Solve the system of linear equations: $$5x + 3y = -3$$

1. **State the problem:** Solve the system of linear equations using elimination: $$5x + 4y = 5$$

1. Fabric Blend Problem: State the problem: We want to find the amounts of cotton, polyester, and nylon in a fabric blend costing 3.25 per pound. Cotton costs 4.00, polyester 3.00,

1. **Problem 2:** Show that $$\sum_{k=1}^n n^{-1} x_k$$ is the same as $$\frac{x_1 + x_2 + \cdots + x_n}{n}$$. Step 1: Write the summation explicitly:

1. **State the problem:** We need to find two numbers $x$ and $y$ such that their sum is 15, i.e., $x + y = 15$, and the product of the square of one number and the cube of the oth

1. The problem states that the graph represents the function $y = a^x$ and passes through the point $(0,1)$. 2. Recall that for any exponential function $y = a^x$, when $x=0$, $y =

1. We are given the exponential function $y = a^x$ and a point on the graph $(2, 64)$. 2. Substitute $x = 2$ and $y = 64$ into the equation:

1. **State the problem:** We need to sketch the graph of the exponential function $$y = 5^x$$ and label the points where it intersects the axes. 2. **Identify key points:** The fun