Then you must include on every digital page view the following attribution: If you are redistributing all or part of this book in a digital format, Then you must include on every physical page the following attribution: If you are redistributing all or part of this book in a print format, Want to cite, share, or modify this book? This book uses the This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission. We will add one more step to solve for y. In the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through the point ( 2, 3 ). m ( x − x 1 ) = y − y 1 Rewrite the equation with the y terms on the left. m ( x − x 1 ) = ( y − y 1 x − x 1 ) ( x − x 1 ) Simplify. y − y 1 = m ( x − x 1 ) m = y − y 1 x − x 1 Multiply both sides of the equation by x − x 1. M = y − y 1 x − x 1 Multiply both sides of the equation by x − x 1. When we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD. We will now see how to solve a rational equation for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. When we solved linear equations, we learned how to solve a formula for a specific variable. Solve a Rational Equation for a Specific Variable That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.Īn algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution to a rational equation.įor rational function, f ( x ) = x − 1 x 2 − 6 x + 5, f ( x ) = x − 1 x 2 − 6 x + 5, ⓐ find the domain of the function ⓑ solve f ( x ) = 4 f ( x ) = 4 ⓒ find the points on the graph at this function value. So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. But because the original equation may have a variable in a denominator, we must be careful that we don’t end up with a solution that would make a denominator equal to zero. Then, we will have an equation that does not contain rational expressions and thus is much easier for us to solve. We will multiply both sides of the equation by the LCD. We will use the same strategy to solve rational equations. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions. We have already solved linear equations that contained fractions. įor more teaching and learning support on Algebra our GCSE maths lessons provide step by step support for all GCSE maths concepts.Rational Expression Rational Equation 1 8 x + 1 2 y + 6 y 2 − 36 1 n − 3 + 1 n + 4 1 8 x + 1 2 = 1 4 y + 6 y 2 − 36 = y + 1 1 n − 3 + 1 n + 4 = 15 n 2 + n − 12 Rational Expression Rational Equation 1 8 x + 1 2 y + 6 y 2 − 36 1 n − 3 + 1 n + 4 1 8 x + 1 2 = 1 4 y + 6 y 2 − 36 = y + 1 1 n − 3 + 1 n + 4 = 15 n 2 + n − 12 Solve Rational Equations Looking forward, students can progress with more solving equations worksheets and to additional algebra worksheets, f or example a factorising worksheet, or a simultaneous equations worksheet. We can work out the number of solutions a quadratic equation has by using the discriminant.Ĭompleting the square can also be used to solve a quadratic equation this method can help to identify the turning points of the quadratic graph produced from the equation. The solutions can be left as surds (with square roots), or written as decimals as required. Here the coefficients of the different terms are substituted into the formula and the solutions are calculated. Then each factor is considered in turn to be equal to zero and the solution is found.Īnother method we can use to solve quadratic equations is using the quadratic formula. After rearranging the equation so that the right hand side is equal to 0, the quadratic expression on the left hand side of the equals sign is factorised so that it is written as a product of two factors. If the equation is more complex and contains a squared term and the linear term we can use a variety of methods to solve it. If there is a single squared variable in the quadratic equation it can be solved by rearranging the equation to put the unknown on one side of the equals sign and all the other terms on the other side. These can be solved using a variety of methods and there are usually two solutions. Quadratic equations contain variables that are raised to a power no higher than two.
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