Chapter 1.1 & 1.2 Explained with Problem Solutions

Welcome to the world of variables and basic algebra!A variable, like x, is a symbol that represents an unknown number.The same variable can represent different values. For example, x could be 3, 7, or 10.Let's look at how we can perform operations with variables.When we know the value of a variable, we can evaluate expressions by substituting that value.We can translate word phrases into algebraic expressions.Now that we understand variables and basic expressions, we're ready to solve equations.The addition property of equality states that we can add or subtract the same value from both sides of an equation.To solve for x, we subtract 5 from both sides to isolate the variable.This gives us our solution: x equals 7.Now let's solve a two-step equation using both the addition and multiplication properties.First, we subtract 5 from both sides to isolate the term with our variable.This gives us 2x equals 8.To solve for x, we divide both sides by 2.This gives us our solution: x equals 4.Let's solve a more complex equation with variables on both sides.First, we subtract x from both sides to get all variable terms on one side.Combining like terms gives us 2x plus 2 equals 10.Subtract 2 from both sides.Finally, divide both sides by 2 to get x equals 4.It's important to always check our solution by substituting it back into the original equation.Let's substitute x equals 4 into our original equation.Simplifying both sides...We get fourteen equals fourteen, confirming our solution is correct.When solving word problems, following a systematic approach is crucial.Let's solve our first problem about John and Mary's ages.We'll let x represent Mary's age, then write an equation based on the problem.Now we can solve for x by subtracting 5 from both sides, then dividing by 2.Let's verify our answer by checking if it makes sense.Now let's solve a problem involving money.We'll let s represent the cost of snacks and write an equation using the total cost.First calculate the cost of tickets, then solve for the cost of snacks.Let's verify our solution by checking the total cost.Finally, let's solve a problem about distance and speed.We can find the average speed by dividing the total distance by the time taken.Let's plug in our values and solve.We can verify our answer by checking if the distance equals speed times time.Remember to always check if your answer makes sense in the context of the problem.