![]() This means that the radius of the circle is approximately \(6.9\) inches. From here we can isolate the variable \(r\) by dividing both sides by \(π\), and then finding the square root of this. When the values are plugged into the equation, \(A=πr^2\) becomes \(150=πr^2\). Let’s use the formula to determine the radius of a circle with an area of \(150\) square inches. As a result, you may have combined the x 3 and x 2 terms. This equation has two variables, \(A\), and \(r\). For questions 16-20, solve the problem and enter your answer in the grid on the answer sheet. The equation is used to find the area of a circle, and it states that \(A=πr^2\). Next, add 2 2 2 to both sides, then subtract 3 x 3x 3x, and finally finish it off by dividing 5 5 5 to both sides. ![]() This equation is often seen in the world of geometry. Let’s look at another example of an equation with multiple variables. Inverse operations are crucial for solving one- and two-step equations. The inverse of adding \(5\) is subtracting \(5\), and the inverse of multiplying by \(8\) is dividing by \(8\). Inverse operations are like opposite operations. “Undoing” operations is referred to as inverse operations. To check that we have solved correctly, take this value for \(y=1\), and plug it back into the equation to see if it is truly balanced: \(6+2(1)=8\). Now the equation shows \(y=\frac\) or \(y=1\). The variable \(y\) is currently multiplied by \(2\), so to “undo” this operation we need to divide both sides by \(2\). Now we are only one step away from knowing the value of \(y\). This needs to be done to both sides of the equation in order to keep it balanced. In order to “undo” a positive \(6\) on the left side of the equation, we need to subtract \(6\). This can be done by “undoing” the operations that are affecting the \(y\). Solving for x means isolating x (or whatever the variable of interest it doesnt have to be called x) on one side of the equal sign (it doesnt matter which). Now the goal is to isolate the variable \(y\). ![]() At this point, there is only one variable in the equation. Now multiply \(3\times2\) so the equation becomes \(6+2y=8\). Then click the button and select Solve for x to compare your answer to Mathways. If the value of \(x\) is \(2\), what will the value of \(y\) be? The first step in this example is to plug in \(2\) for \(x\). Once youve completed all the questions (assuming youve got some time. The equation \(3x+2y=8\) contains two variables. Let’s look at a few examples of equations with more than one variable. However, some equations will have more than one variable. Many equations will have only one variable, as in the previous example. For example, in the equation \(10x+30=90\), the solution for \(x\) is \(6\) because when \(6\) is multiplied by \(10\), and then added to \(30\), the result is \(90\), creating a balanced equation. The solution of an equation is a value that makes the equation balanced. ![]() When the variable is on its own, it reveals the solution. The main objective when solving an equation is to isolate the variable. Step 3: Isolate the variable using inverse operations. Step 2: Move all of the parts containing the variable you are solving for to the same side of the equation. There is no value that will ever satisfy this type of equation.Step 1: Simplify both sides of the equation. This type of equation is never true, no matter what we replace the variable with. The last type of equation is known as a contradiction, which is also known as a No Solution Equation. For this type of equation, the solution is all real numbers. The solver will then show you the steps to help you learn how to solve it on your own. No matter what value we replace x with, the equation is true. To solve your equation using the Equation Solver, type in your equation like x+45. If we simplified each side we would get: 3x - 15 = 3x - 15. The left and the right side can be simplified to match each other. The second equation, an identity is always true, no matter what value replaces the variable. Answer the Questions in the spaces provided 5 Solve 5 m 12 (Total for Question 2 is 1 mark) g 6 Solve 5g 40 (Total for Question 6 is 1 mark) x 2. This equation is true when x = 4, but false when x is any other value. As an example, suppose we look at 3x = 12. The first type of equation, known as a conditional equation is true under certain conditions, but false under others. These are conditional equations, identities, and contradictions. When solving equations, we will encounter three types of equations. When we encounter special case equations, we will see No Solution Equations and Equations that have infinitely many solutions. In this section, we learn about special case linear equations.
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